LIBRARY 

.  OF  THE 

University  of  California. 


GIF^T  OF^ 


A 

KEY 

,  CONTAINING 

THE  STATEMENTS  AND  SOLUTIONS  OF  QUESTIONS 
PEOFESSOE   OHAELES   DAVIES' 

NEW  ELEMENTARY  ALGEBRA. 

FOE    THE 

USE   OF   TEACHERS   ONLY. 

V     or  THF 

UNIVER  « 


NEW  YORK  : 

BARNES  &  BURR,  PUBLISHERS,  51  &  53  JOHN  ST. 

CHICAGO  :    GEORGE  SHERWOOD,  118  LAKE  ST. 

CINCINNATI:     RICKEY     <fe    CARROLL. 

ST.  LOUIS :   KEITH  <fc  WOODS. 

1863. 


Entebed,  according  to  Act  of  Congress,  in  the  year  Eighteen  Hundred  and  Fifty-nine, 

By    CHARLES    DAYIEB, 

In  the  Clerk's  Office  of  the  District  Court  of  the  Southern  District  of  New  York. 


"William  Denyse, 

Stereotype!'  and  Electrotyper^ 

188  William  Street,  N.  Y. 


Wood,  Printer^ 
Cor.  Fulton  and  Dutch  Sts.,  N.  Y, 


P  E  E  F  A  C  E. 


The  question,  whether  a  Key  to  a  work  on  Mathematics 
facilitates  the  acquisition  of  knowledge,  is  one  about  which 
there  is  much  diversity  of  opinion.  If  the  business  of 
teaching  were  pursued  as  a  profession — if  the  teachers  in 
our  schools  and  seminaries  looked  to  no  other  employment, 
and  gave  their  entire  thoughts  and  time  to  the  business  of 
instruction,  they  would  have  abundant  means  to  prepare,  in 
the  best  manner,  all  the  exercises  for  their  pupils. 

But,  as  yet,  the  case  is  far  different.  Teaching,  with  most 
instructers,  is  an  occasional  and  temporary  business,  and  not 
a  permanent  profession.  Engaged,  generally,  in  prejDaring 
themselves  for  other  pursuits,  and  at  the  same  time  giving 
instruction  in  various  branches  of  education,  they  have  nei- 
ther the  time  nor  opportunity  for  that  careful  preparation 
which  is  needed,  and  must,  therefore,  avail  themselves  of 
all  the  aids  which  they  can  command. 

It  was  not  intended,  originally,  to  prepare  a  Key  to  the 
Elementary  Algebra,  but  the  urgent  request  of  many  teach- 
ers has  changed  that  determination. 

iii 


IV  PREFACE. 

It  was  not  thought  best  to  work  out  the  simple  examples 
which  are  given  as  illustrations,  nor  those  which  are  given 
to  perfect  the  scholar  in  the  meclianical  part  of  Algebra ;  and 
hence,  the  work  in  the  Key  is  limited  to  the  problems,  with 
the  exception  of  two  examples,  one  on  page  114,  and  the 
other  on  page  221.  The  problems,  it  was  supposed,  pre- 
sented difficulties  in  the  statements  only,  which  are  fully 
given,  leaving  the  solution  of  the  equations  to  be  made  by 
the  pupil.  This  will  obviate  much  of  the  misuse  to  which  a 
Key  may  be  applied,  should  it  chance  to  fall  into  the  hands 
of  the  student. 

The  large  figures  at  the  head  of  each  page  point  out  the 
corresponding  page  of  the  Algebra. 


KEY 

TO 

DAYIES'  NEW  ELEMENTARY  ALGEBKA. 


Pag^e  114. 

(25.) 


(a  +  5)  (a;  ^  b)  _  3^  ^  4Mj--J^  _  ^^       a?  -  hx 
a  —  b  a  +  b  b 

Multiplying  by  the  least  common  denominator, 

{a  —  b)  («•+  b)b^  and  canceling,  we  have, 

{a  +  by  {x  -  b)b  -  Sab{a  -  b)  {a  +  b)    =   {iab  -  ¥) 
{a  —  b)b  -  2x{a  —  b)  {a  +  b)b  +  {a^  -  bx)  {a  -  b)  {a  +b). 

Performing  indicated  operations,  we  have, 

a^bx  +  2a52aj  +  b^x  -  aW  -  2ab'^  —  5^  -  da^b  +  Sab^  = 
4aW  -  ctb^  -  4cab^  +  b^  -  2a^bx  +  2b^x  +  a*  —  a^bx 

By  transposing, 
a'^bx  +  2ab'^x  +  ¥x  +  2a2^>a;  —  2b^x  +  a^^a;  -  5^^  = 
4a252  -  a63  -  4aP  +   )*  +  «*  -  «^^^  +  a^b^  +  2ab^  -{- 
b'  -f  3a36  —  3a53. 


6  KEY  TO  DAVIES'  NEW  ELEMENTAEY  ALGEBEA.        [125. 

Collecting  terms  and  factoring, 

2bx{2a'^  +  ab  ^  P)  z=  a^  +  Sa^b  +  4a^b^  -  eaP  +  25*; 
a^  +  Sa^b  +  ia^^  -  6aP  +  25* 


hence, 


25(2a2  +  ab  -  b^) 


Page   125. 

(13.) 

Denote  D'^s  share  by  ic.    Then,  by  the  conditions  of  the 
question, 

aj  +  360  =  B's  share, 

and  2x  +  720  —  1000  =  A^s  share. 

But,  X  z=:  D^s  share, 

and  360  —    C'>s  share; 

hence,  4a;  +  440  =  2520,  the  whole  estate;  from 

which  equation  we  find  cc  =  520. 

(14.) 

Let  X  denote  the  share  of  each  daughter.     Then,  by  the 
conditions  of  the  question, 

2x  =  the  share  of  each  son. 
Also,  since  there  are  three  daughters  and  two  sons, 
Sx  =z  the  amount  received  by  the  daughters, 
and  4x  =  the  amount  received  by  the  sons ; 

also,  7cc  +  500  =  the  amount  received  by  the  widow, 
and  14aj  -{-  500  =  7500,  the  whole  estate ;  and  from 
this  last  equation  the  value  of  cc  is  readily  found,  equal  to  500. 

(15.) 

Let  X  denote  the  number  of  women.    Then,  by  the  con- 
ditions of  the  question, 

X  +  S  =:  the  number  of  men. 


126.]  EQUATIONS  OF  THE  FIRST  DKGKEE.  7 

and  2aj  +  8  +  20   =   children. 

But,  X  =z  women ;  hence, 

4aj  +  36  =  180,  the  whole  number  ;  from 
which  we  find  cc  =  36. 

(16.) 
Let  X  denote  the  share  of  the  youngest  brother. 
Then,  x  +    40  =  2d    son's  share, 

X  -{-    80  =:  3d    son's  share, 
aj  +  120  =z  4th  son's  share, 
cc  +  160  =  5th  son's  share, 
and  5x  +  400  =  2000,   the  whole  estat^  ;  from 

which  we  find  x  =  320. 

(17.) 

Let  the  share  of  A  be  denoted  by  x.  Then,  since  A^s 
share  is  to  be  to  JB^s  as  6  to  11,  it  follows  that  J5'5  shar<? 
will  be  -y-  of  u4'5  ;  hence, 

-—X  =  j5'5    share,' 

and  X  +  —x  +  300  =   C^s  share ; 

22 
hence,  2x  -^  —x  +  300  =  2850,  the  whole  estate; 

6 

from  which  equation  we  find  x  =  450. 

(18.) 

Let  X  denote  the  number  of  paces  taken  by  the  first 
person,  fi^om  the  time  of  starting  till  the  distance  between 
them  is  300  feet.  Then,  the  number  of  paces  taken  by  the 
second  will  be  represented  by  5x.     But  since  the  paces  of 


8  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.        [126. 

the  first  are  3  feet,  and  those  of  the  second  1 J  or  -^  feet,  the 
distances  traveled  will  be 

Sx  =  the  distcmcG  traveled  by  first, 

and       --  X  bx  =z  —x  ~  the  distance  traveled  by  second ; 

15 
hence,  -—x  —  ^x  —  300,  their  distance  apart ;  from  which 

we  find  X  =  6Q^  ;  that  is,  the  person  who  steps  the  longest 
will  have  made  66|  paces ;  and  since  each  pace  is  3  feet,  he 
will  have  traveled    • 

66|-  X  3   =  200  feet. 

15 
If,  instead  of  subtracting  Sx  from  — a?,  we  had  written 

o 

the  equation, 

15 

3a?  —  —-cc  =  300, 

we  should  have  found, 

X  =   -  66f, 

which  would  have  shown  that  the  second  person  traveled 
farther  than  the  first ;  which  is  indeed  proved,  when  the 
distance  traveled  by  the  second,  minus  the  distance  traveled 
by  the  first,  is  positive. 

(19.) 

Let  X  denote  the  number  of  days  which  they  worked. 

Then,      2x  =z  the  number  of  dollars  earned  by  cai-penters, 

24cc 

-— -  =  12a;,  the  amount  earned  by  journeymen, 

o 

-a;  =     2a:,  the  amount  earned  by  the  apprentices ; 

hence,  16a;  =:  144,  the  whole  sum  earned ;  Vhence  we  find 
cc  =  9. 


126.1  EQUATIONS  OF  THE  FJKST  DEGREE.  9 

(2O0 

Let  the  sum  at  interest  be  denoted  by  «j. 

4 
Then,     -x  =  what  bears  an  interest  of  4  per  cent., 

and         -cc  =  what  draws  5  per  cent. 
o 

Then,  since  the  interest  which  accrues  on  any  sum  for  a 
year,  is  equal  to  the  sum  multiplied  by  the  rate,  divided  by 
100,  we  shall  have, 

4  4  4 

-X  X  — —  =  T^^t  what  the  first  produced, 

15  1 

and  ~x  X  -— -  =  - — X.  what  the  second  produced  : 

5  100    100  '  i'       J 

then,   A«,  +  _L«,=,2940; 

or,    400a;  +  125a;  =  36750000  ;  or,  a;  =  '70000. 

(21.) 

There  are  several  ways  in  which  this  example  may  be 
solved : 

First.  We  see  by  the  conditions,  that  the  first  cock  will 
discharge  one  gallon  in  a  minute,  the  second  cock  half  a 
gallon  in  a  minute,  and  the  third  cock  one-third  of  a  gallon 
in  a  minute.  Hence,  the  quantity  discharged  by  the  three 
cocks  in  a  minute,  is  equal  to 

1  +::;  +  ::  =  -^     2jallons. 

Then,       —  gals.   :   60  gals.     :  :     1  m.   :    32y\  minutes. 

But,  to  resolve  the  question  in  a  general  way,  mthout 
regard  to  the  contents  of  the  cask,  let  x  denote  the  part  of 
1* 


10  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.        [126. 

the  cask  which  would  be  emptied  in  a  single  minute.  Then, 
as  the  first  cock  would  empty  q\  of  the  cask,  the  second 
•j-^o  of  it,  and  the  third  jl^  of  it,  in  one  minute,  we  have, 


X 

= 

1 

60 

+ 

1 

120 

+ 

1 

180' 

X 

= 

6 
360 

+ 

3 
360 

+ 

2 
360' 

X 

= 

11 

360 

• 

or, 


or, 

Now,  as  JgV  of  the  cask  is  emptied  in  one  minute,  it  is 
evident  that  to  empty  the  entire  cask  will  require  as  many 
minutes  as  the  number  1  contains  ^lu  ;  that  is,  32y8y. 

(22.) 
Let  the  number  of  trees  in  the  orchard  be  denoted  by  a. 

Then,  ^        o^  =  apple  trees, 

-X  =  peach  trees, 

-X  =z  plum  trees, 

20  =  cherry  trees, 
80  =  pear  trees, 

and  X  =z  -x  -^  -X  +  -X  +  120  +  80: 

2  4  6 

from  which  we  find  x  =  2400. 

(23.) 

Let  the  number  of  sheep  be  denoted  by  x.  Then,  by 
observing  the  statement  of  the  last  problem,  we  have, 

^  "^  4^  "*"  6^  "^  8^  "^  12^  "^  ^^^' 
which,  gives,  x  =  1200. 


127.]  EQUATIONS  OF  THE  FIRST  DEGREE.  11 

(24.) 

Let  X  =  the  value  of  the  horse. 

Then,  —  =  the  value  of  the  saddle, 

and,  ^  "•"  10  "^  -^^^^ 

which  gives,  a;  =  120. 

(25.) 
Let    X    denote  the  amount  of  rent  last  year. 

Sx 
Then,    —    equal  the  excess  of  the  present  over  the  past 

year,  and  the  rent  of  the  present  year  will  be  expressed  by 
.  +  -   =   1890; 


which  gives, 

X   =:    1750. 

(26.) 

Let 

X  =  the  number. 

Then, 

X  —  5  =  the  difference,  and 

?  X  (aj-5)  =  40; 

or 

2a;  —  10  =  120;    or,    x  =  61 

(27.) 

Let 

X  =  the  length  of  the  post. 

Then, 

'^x  +  h^+10  =  X, 

from  which  ' 

we  have,         x  =  24. 

12  KSY  TO   DAVIES'  NEW  ELEMENTARY  ALGEBRA.      [127 

(2§.) 
Let  X  =  the  number. 

Then,  x  --  -x  — -x  =  66 ; 

4  5 

from  which  we  have,       x  =  120. 

(29.) 

Let  the  number  of  beggars  be  denoted  by  x. 
Then,  by  the  conditions, 

3a;  —  8  =  the  amount  of  money  he  had ; 
also,         2cc  +  3  =  the  amount  of  money. 
Hence,     3a;  —  8  =  2a;  +  3,   or  a;  =  11. 

(30.) 
Let      X  =  the  number  of  shillings  which  he  had. 
Then,    -x  =  what  he  first  lost ; 

and,    a;  —  -a;  +  3  =  what  he  had  left  after  borrowing 

Zx  -f-  3        Sx  -f-  12 
Then,    - — - —  =  — -— —  =  what  he  lost  the  second  time ; 

o  12 

and  this,  taken  from  Ja;  +  3,  or  what  he  had  at  the  com- 
mencement of  the  second  game,  will  give  what  he  had  kA; ; 
that  is, 

3      ,      ^         3aj  +  12 

from  which  we  find,  x  —  20s, 


128.]  EQUATIONS  OF  THE  FIRST  DEGEEE.  13 

(31.) 
Let    X    denote  the  amount  laid  out  by  each. 
Then,        cc  +  126   =  what  A  had,  after  gaining, 
and,  X  —    87  =F  what  B  had,  after  losmg. 

Then,         aj  -f  126   ■=::  2{x  -  87)  =  2iB  -  174; 
from  which  we  find,         x  =  300. 

(32.) 

Let    X    denote  the  sum  which  he  had  at  first. 
Then,    x  —    2  =  what  he  had  after  spending ; 
2 


2aj  —    4  =  what  he  had  after  borrowing, 
and,     2a3  —    6  =  what  he  had  after  spending  at  second 
2  tavern ; 


403  —  12   =  what  he  had  after  borrowing, 
and,     4a;  —  14  =  what  he  had  after  spending  at  third 
2  tavern ;  ' 


8a;  — •  28,  what  he  had  after  borrowing,  and 
then,  8a;  —  305.  =  0. 

Hence,  a;  =  3f5.  =  35.  9 c?. 

(33.) 

Let    X    denote  the  number  of  yards  in  the  piece. 
Then,  3a;  —  57  =  what  remained  of  the  first  3  pieces, 

and,  a;  —  17  =  what  remained  of  the  fourth  piece. 


Hence,  4a;  —  74  =  142  yards,  what  remained  in  all. 

Therefore,     4a;  =  142  +  74 ;    or,   x  —  54. 


14  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.       [128. 

(34.) 
Let  the  number  of  cavany  soldiers  be  denoted  by  x. 

Then,    Sec  =  the  number  of  the  artillery,  and    9a;  =  the 
number  of  the  infantry  :  also, 

a;  +  3a;  +  9a;  =  2600 ;    or,    x  =  200. 

(35.) 

Let  X  denote  the  number  of  miles  he  traveled  on  horse- 
back. 

Then,    3Ja;    will  denote  the  distance  traveled  by  water, 

no 

and    3|a;  x  2J  =  —  a?  =  what  he  traveled  on  foot. 

63 
Consequently,     x  +  S^x  +  -—a;  =  2970;    x  =  240. 

o 

(36.) 
Let     X    denote  the  number  of  pieces  JB  received. 
Then,  -x   will  denote  the  number  A  received;  and, 

o 

5  8a;        ^,  . 

a;  +  -a;  =  -—  =  the  entire  sum. 

o  o 

But,  by  the  conditions, 

5         5      „   8  5         40 

-a;--    of   3^  =  3«^~27^  =  50; 

that  is, 

45         40  «.^  5  ^^ 

27^  -  27^  "=  ^0;    ^^^    27^  =  50;    or,    x  =  270. 

Hence,  A's  share  is  450  pieces. 


129.]  EQUATIONS  OF  THE  FIRST  DEGREE.  15 

(37.) 

Let  the  first  number  be  denoted  by  x ;  then,  the  second 
will  be  denoted  by  mx ;   and  the  third  by   nx. 

Hence,  x  +  mx  +  7ix  =  «,    which  gives, 

a  ^,  an  ^^  am 

1st,   X  =:  - — I — ;   2d,   — — j — ;   3d, 


I  -\-  771  -\-  n  1  -\-  m  -\-  n  \  -\-  m,  -\-  n 

(38.) 

Let   X    denote  the  number  of  dollars  in  the  younger's 
portion. 

Then,         x  -\-  -x  =  the  portion  of  the  second ; 

o 

and,  2x  =  the  portion  of  the  thu'd. 

By  the  conditions, 

^  +  ^  +  Q^  +  2a3  =  1170; 

o 

that  is, 

3£C  +  2aj  +  cc  +  6aj  =  3510,    and,    x  =  270. 

(  39.) 

Let    X    denote  the  number  to  be  furnished  by  the  first. 

Then,  ~x  =  that  of  the  second. 

o 

7     5     35 
and,         -  of  -a;  =  —x  =   that  of  the  third, 

5    35 
and,         X  +  ~x  +  —x  =   594 ; 

that  is, 

72aj  +  120CC  +  105a;  z=  42768,    and,    x  =  144. 

(40.) 
Let    X    denote  the  number  of  dollars  in  A^s  portion. 


16  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.       [129 

Then,  x  =  A'^s  portion,  in  dollars ; 

and,  2x  +  200  =  B's, 

^x  —'400  =    C's, 

2^X  +      50     r:=    J9'5, 

and, 

Q1.X  150 

8Jaj  —    150  +  -^ — h    475  =     5600,    the  estate; 

or, 

17 
Ux  —    600  +  —X  —  150  +  1900   =:   22400; 

or, 

68a;  —  1200  +  17a;  —  300  +  3800  =  44800, 

hence,     85a;  =  42500;  and  x  =   500. 

(41.) 

Let    x    denote  the  number  of  quarts  that  will  fill  the  first 

cask. 

4  3 

Then,    x  —  -x  =z  ro;  =  the  contents  of  the  second : 

7  7  ' 


3 


—  -l-a;l  ::=  —X  =  contents  of  third  cask; 


9  9 

and,        — X  =z  —  ,    the  contents  of  fourth  ; 

or,  contents  of  fourth,     =  —a;  =  -x. 

28  7 

rw.,  '94 

Then,  a;  =  — a;  +  -a;  +  15  ; 

or,  28a;  =  9a;  +  16a;  +  420, 

and,  X  =   140. 


129.]  EQUATIONS  OF  THE  FIRST  DEGREE.  17 

(42.) 

Let  the  number  of  days  be  denoted  by  x. 

Then,  since  the  first  courier  travels  four  miles  a  day,  we 
have, 

4a;  +  10  X  4  =  453  +  40  =:  the  distance  traveled  by  the 

first  courier ;  and 
9cc  =  the  distance  traveled  by  the  second  ; 

hence,  %x  —  4a3  +  40 ;    or,    cc  =  8. 

(43.) 

Rate  of  first  courier,    31i~-5  =  6.3  miles. 
Rate  of  second   "         22^  -r-  3   =  7.5  miles. 

8    X  6.3         50.4         ^^_ 

X  —  — T—   =  — — T-  =  42  hours. 

7.5  —  6.3  1.2 

(44.) 

The  first  will  have  traveled  3^  x  8  =  28  miles,  at  the 
time  the  second  departs;  hence  they  will  be  52  miles  apart. 
N"ow,  if  we  denote  by  x  the  number  of  hours  after  the 
departure  of  the  second,  until  they  meet,  we  shall  have, 

Z\x  ~  distance  traveled  by  the  first,  after  the  second  starts ; 

and, 

h\x  =z  the  distance  traveled  by  the  second ; 

hence,  S^x  +  5}x  =  52 ; 

'^  31 

or,  -X  -4 X  =  52  ; 

'  2.6  ' 

42  +  62a;  =  624 ;    or,    x  =  Q  hours. 


18  KEY  TO  DAVIES'  NEW  ELEMFNTARY  ALGEBRA.       [135. 

(120 

Let  the  aumber  of  dollars  possessed  by  A^  be  denoted 
by  cc,  and  i?'5  number,  by  y. 

Then,  by  the  first  condition, 

X  +  40  =  5(y  —  40)   =  5^  —  200 ; 

by  the  second,  cc  +  y  =  120  ;  from  wWch  we  readily  find, 

X  =  60,     and    y  =  60. 

(13.) 

Let  X  denote  the  humber  of  years  in  the  father's  age. 

Then  the  father's  age,  twenty  years  before,  will  be  re- 
presented by  X  —  20,  and  that  of  the  son,  by  y  —  20. 
Hence,  by  the  first  condition, 

a;  —  20  =:  4(2/  -  20)   :=   4y  —  80 ; 

by  the  second,  x  =z  2y  ;    from  which  we  find, 

X  =  60,     and     y  —  30. 

(14.) 

Let  X  denote  the  number  of  dollars  which  the  elder  had, 
and  y  the  number  which  the  younger  had;  then,  by  the 
first  condition, 

aj  —  -aj  =  2/  4-  1000 ; 

and,  by  second  condition, 

a;  —  iaj  +  2000  =  2{y  +  1000  —  500); 

or,  4aj  —    aj  +  8000  —  Sy  -{'  8000  —  4000  ; 

or,  3a;  =  8?/  —  4000; 

from  which  the  values  of  x  and  y  are  easily  found. 


137.]  EQUATIONS  OF  THE  FIRST  DEGREE.  19 

I 

(15.) 

Let  X  denote  the  number  of  apples  which  John  had,  and 
y  the  number  Charles  had. 

Then,  x  -  \f>   ==  2/  -f  15, 

and  03  +  15  =:  15(2/  -  15)  -  10; 

from  which  we  have  x  and  y. 

( 16.)      • 
Let  X  denote  the  number  of  dollars  in  A^s  salary,  and  y 
that  in  ^'5  ;  then, 

aj  +  y  r=  900 ; 

and  by  the  second  condition, 

a,  -  !«,  =  y  +  ia,; 
from  which  x  and  y  are  easily  found. 

Page   137, 

(11.) 

Let  X  denote  the  number  of  dollars  which  A  has,  and  y 
the  number  B  has ;  then, 

by  the  first  condition,        aj  +  2/  =  20000  ; 
by  the  second  condition,  y  =:  Sx; 

whence,   x  z=  5000,   and  y  =  15000. 

(12.) 

Let  X   denote  the  value  of  the  first,  and  y  that  of  the 
second. 

Then,  by  the  first  condition,         x  -{-  1  =  Sy; 

and,  by  the  second  condition,    5x  =  2/  +  V  ; 

from  which  we  have  x  and  y. 


20  KEY  TO  DAVIES'  NKW  ELEMENTARY  ALGEBRA.       [146. 

(13.) 
Let  the  numbers  be  denoted  by  x  and  y. 
Then,  by  1st  condition,  6cc  =  Sy, 

and  by  the  second,  aj  —  1   =  y  —  2  ; 

from  which  we  have,      a;  =  5,     and    y  z=z  ^. 

(14.) 
Let  the  numbers  be  denoted  by  x  and  y. 


By  first  condition, 

aj  +  2   = 

Hy; 

or, 

a;  +  2   = 

13 

by  second  condition. 

X 

2   ~" 

2/  +  4; 

from  which  we  find, 

X  =z  24, 

and    2/  =  8. 

(15.) 

Let  the  present  ages  of  the  father  and  son  be  denoted  by 
X  and  y;  then, 

by  1st  condition,  a;  —  12  =  2y ; 

by  2d  condition,         4{y  —  12)  +  12  =z  a;  +  12 ; 

from  which  we  have,     x  =  12^    and    y  =  30. 

Page  145. 

(4.) 
Let  X  and  y  denote  the  numbers. 
Then,  x  —  y  z=  *Jy 

and  a  4-  2/  =  ^^ ; 

which  equations  give,     aj  =r  20,     and    y  =  13. 


146.]  EQUATIONS  OF  THE  FIRST  DEGREE.  21 

(5.) 
Let  X  denote  the  greater,  and  y  the  lesser  part. 
Then,  by  1st  condition,        x  ■\-  y  =  75  ; 
and  by  2d  condition, ,  Sx  =  1y  +  15; 

which  give,  aj  =  54,     and    y  =  21. 

(6.) 

Let  X  denote  the  number  of  gallons  of  wine,  and  y  the 

cider. 

X  ~\~  1/ 
Then,  by  1st  condition,        — -— ^  +  25   =  x; 

and  by  2d  condition,  — — ^  —    5=2/; 

from  which  we  have,        x  =  85,     and    y  =  35. 

( ^0 

Let  X  denote  the  number  of  guineas,  and  y  the  number 
of  moidores  used. 

N^ow,  it  is  evident  that  the  number  of  pieces  used,  of  each 
kind,  multiplied  by  the  number  of  shillings  in  the  jDiece,  will 
give  the  number  of  shillings  paid  in  that  particular  kind  of 
money.  That  is,  21i«  will  be  the  number  of  shillings  paid 
in  guineas,  and  21  y  the  number  paid  in  moidores.  Then, 
observing  that  the  whole  bill,  £120  =  2400^.,  we  have, 
by  1st  condition,  x  +  y  =  100; 

by  2d  condition,  21CC  +  21y  =  2400; 

which  give,  a;  =  50,     and    y  =  50, 

(8.) 

Let  X  denote  the  distance  traveled  by  the  first,  and  y 
the  distance  traveled  by  the  second. 
Then,  x  +  y  =  150 


22  KEY  TO  DA  VI  is' m:w  i;lkmkntary  algkbra.      [146. 

But,  since  the  first  travels  8  miles,  while  the  second  travels 
but  7,  the  distance  which  they  respectively  travel  will  be  in 
the  proportion  of  8  to  V  ;  that  is, 

ce  :  2/  :  :  8  :  7; 
or,  1x  =  ^y\ 

from  which  we  find  x  =  80,  and  y  =  VO;  and  if  the  entire 
distance  traveled  by  each  be  divided  by  the  distance  trav 
eled  each  day,  the  quotient  AviU  be  the  time,  10  days. 

(0-) 

Let  X  denote  the  number  of  votes  cast  for  the  first,  and 
2/  the  number  cast  for  the  second. 
Then,  x  -{-  y  —  375, 

and  X  —  y  =z     91  ; 

which  give,         x  =  233,     and    y  =  142. 

(lO.) 

Let  X  denote  the  value  of  the  poorest  horse,  and  y  that 
of  the  other. 

Then,  by  1st  condition,         aj  +  50  =  2i/, 

and  by  2d  condition,  y  -\-  50  =  Sx; 

which  give,  x  =  £30,     and    y  =  £40. 

(XI.) 

In  this  example,  we  must  bear  in  mind  that  the  minute 
hand  goes  entirely  round  the  face  of  the  clock,  while  the 
hour  hand  passes  from  one  hour  to  the  other ;  that  is,  the 
minute  hand  travels  twelve  times  as  fast  as  the  hour  hand. 

If,  then,  we  suppose  the  face  of  the  clock  to  be  divided 
into  twelve  equal  parts,  corresponding  to  the  hours,  and  x 
and  y  to  represent  the  distances  passed  over  by  the  hour 


146.]  EQUATIONS  OF  THE  FIRST  DEGREE.  23 

and  minute  hands,  from  the  time  of  separating  mitil  they 
are  again  together,  we  shall  have, 

and  2/  ~  ^  =   12  ; 

since,  when  the  hands  come  together,  the  minute  hand  will 

have  gained   the  entire  twelve  spaces  on  the   hour  hand. 

Multiplying  the  second  equation  by  12,  and  adding  them 

together,  w^e  have, 

122/  =  2/  +  144; 

144 
or,  y  =  —    =   13Jy; 

that  is,  the  minute  hand  will  have  gone  once  around  the  face, 
and  lyV  ^^  ^^  hour  spaces  in  addition ;  consequently,  the 
time  required  will  be  1  hour,  5  minutes,  Jy  of  5  minutes,  or 
y\  of  one  minute. 

If  we  subtract  the  second  from  the  first  equation  of  con- 
dition, we  have, 

12 
\\x  =  12,    and    x  =  —  =  1tt5 

that  is,  X  is  equal  to  1  and  yV  of  the  hour  spaces,  which, 
reduced  to  ti7ne^  gives  1  hour  5/y  minutes,  as  before. 

(12.) 

Denote  by  x  the  portion  of  the  beer  which  the  man  would 
drink  in  a  single  day. 

Then,  by  the  conditions  of  the  question,  the  man  and 
woman  together  would  drink  J^  of  the  cask  in  a  single  day, 
and  the  woman  gV  of  it ;  hence,  what  the  man  would  drink 
must  be  equal  to  the  difference ;  that  is, 

1  1  30  —  12  18  1 


X  = 


12        30  360  360         20' 


24  KEY  TO  DAVILS'  NKW  j;LI:MKNTAKY  ALGEBRA.       [146. 

that  is,  the  man  will  drink  gV  of  the  beer  in  a  single  day ; 
and  hence,  the  whole  of  it  in  20  days. 

(13.) 

Let  the  fresh  water  to  be  added  be  denoted  by  x.  Then, 
the  amount  of  the  mixture  will  be  denoted  by  cc  +  32.  But 
the  addition  of  the  fresh  water  will  not  increase  the  quantity 
of  salt  in  the  32  lbs.  of  salt  water;  hence,  the  cc  +  32 
pounds  of  the  mixture  will  contain  one  pound,  or  16  ounces, 
of  salt.  But,  by  the  conditions  of  the  question,  32  lbs.  of 
this  mixture  are  to  contain  2  ounces  of  salt ; 

hence,  a?  +  32  :  32  :  :  16o^.  :  2o2. ; 

consequently,  2cc  +  64  =  512  ; 

or,  X  z=z  224. 

(14.) 

In  this  example,  we  must  bear  in  mind  that,  if  the  rate  of 
interest  be  divided  by  100,  and  the  quotient  multiplied  by 
the  principal,  the  product  will  always  be  the  amount  of 
interest. 

Let  X  denote  the  greater  part,  and  y  the  lesser. 

Then,  x  -\-  y  =:  100000; 

also,  — -  =  what  the  larger  part  produced ; 

4v 
and,  -~  =  interest  of  lesser  part ; 

5x  4?y 

consequently,         -— -  +  —~  =  4640 ; 
^         •^'  100         100  ' 

from  which  we  find, 

X  =  64000,    and    y  =  36000. 


147.]  EQUATIONS  OF  THE  FIRST  DEGREE.  25 

(15.) 

Denote  the  number  of  votes  received  by  the  successful 
candidate,  by  x,  and  the  number  received  by  the  other,  by 
2/.     Then,  by  the  first  condition, 

X  —  y  =  1500. 

Had  the  first  received  ^  of  y  in  addition,  the  second 
would  have  received    y  —  \y  =z  f  y,  and  we  should  have, 

a;  +  -y  zzr  3  X  -2/-  3500; 

or  4a3  4-  2/  =  9^  —  14000; 

from  which  we  have, 

X  =  6500,    and    y  =  5000. 

(16.) 

Let  X  denote  the  value  of  the  gold  watch,  and  y  that  of 
the  silver  watch. 

Then,  x  +  25  =  S^y  =  \j, 

2/+  25  =.  1+  15; 

that  is,  2cc  +  50  =  72/, 

and  22/  +  50  =  cc  +  30 ; 

from  which  we  have, 

X  =   80,    and    y  =  30. 

(ly.)  ^ 

The  separate  figures  which  are  placed  by  the  side  of  each 
other,  in  order  to  express  any  number,  are  called  digits. 
Now,  from  the  relative  value  of  these  figures,  resulting  from 
the  places  which  they  occupy,  we  can  easily  see  how  the 
numbers  may  be  expressed.    For  example,  if  the  number  is 

>       or   THF 

UNIVERSITY 

or 


26  KEY  TO  DA  vies'  NEW  ELEMENTARY  ALGEBRA.       [147, 

expressed  by  two  digits,  then  the  first  figure  on  the  right, 
plus  ten  times  the  second  figure,  will  always  give  the  num- 
ber.   Thus, 

36  =  3  X  10  4-  6;    and,    87  =  8  X  10  +  7,    &c. 

If  the  number  is  expressed  by  three  figures,  then  one  hun- 
dred times  the  left-hand  figure,  plus  ten  times  the  middle 
figure,  plus  the  right-hand  figure,  will  express  the  number. 
Thus, 
246   =   100  X  2  +  10  X  4  H-  6   =   200  -f  40  +  6   =   246. 

Let  X  z=z  the  left-hand  digit,  and  y  =  the  other. 
Then,  a;  +    y  =  11 ; 

also,  a;  -f  13  =  3y; 

from  which  we  have, 

a;  =  5,    and    y  z=  Q, 

(18.) 

Let  X  denote  the  number  of  gentlemen,  and  y  the  num- 
ber of  ladies.  Then,  y  —  15  =  the  ladies  who  remained, 
and  X  ~  45  =z  the  gentleman  who  remained.  And,  by 
the  conditions  of  the  question, 

aj  =  2(?/  -  15)   =  2y  -  30, 
and,    5{x  -  45)   =  2/  -  15  ;    or,    5aj  -  225   =  y  -  15  ; 
from  which  we  have, 

X  =  50,    and    y  =z  40. 

.( 10-) 
aiicij^^^if,  X  denote  the  value  of  the  horse  in  dollars,  and  y  the 
number  of  tickets.    If  he  sells  the  tickets  at  $2,  he  will 
receive  $21/ ;  if  at  $3,  he  will  receive  $Sy, 
Then,  2y  =  x  —  30, 

and,  3y  =  aj  +  30 ; 

which  give,  aj  =   150,    and    y  =  60. 


148.]  EQUATIONS  OF  THE  FIEST  DEOEEE.  27 

(20.) 

Let  X  denote  the  number  of  bushels  of  wheat  purchased, 
and  y  the  number  of  bushels  of  rye. 

Then,  lOOoj  +  I5i/  =  11V50  cents; 

also,  100  X  tCC  +  75  X  -2/  =     2750  cents; 

4  5 

or,  25a;  +  15y  =     2750  cents; 

from  which  we  have, 

X  =  80,    and    y  =  50, 

(21.) 

Let  X  denote  the  number  A  took,  and  y  the  number  JS 
took. 

Then,  52  —  cc  =  what  A  left, 

and,  52  —  2/  ==  what  JB  left. 

But,  by  the  conditions, 

X  =  2(52  —  2/),  and    y  =  7(52  —  x)  ; 

that  is,         X  =  104  —  27,  and    y  =  364  —  Ix, 

Hence,        a;  =  48,  and    y  =  28. 

(22.) 

Let  X  denote  the  number  of  dollars  which  A  has,  and  y 
the  number  jB  has. 

Then,        x  +  %y  z=z  1200,    and    y  -^  ^x  —  1200; 

from  which  we  find, 

X  =  800,    and    y  =  600. 


28  KEY  TO  DA  vies'  NEW  ELEMENTARY  ALGEBRA.       [156. 

(23.) 

Let  X  denote  the  price  of  the  brandy,  and  y  that  of  the 
sherry. 

If,  now,  we  make  the  first  mixture,  that  is,  two  dozen  of 
sherry  and  one  dozen  of  brandy,  the  mixture  itself  will 
contain  three  dozens,  and  will,  consequently,  be  worth 
785.  X  3  =  2345.     Hence,  we  have, 

2aj  +    2/  =  234,    for  the  first,  and, 
7cc  +  2y  =     79  X  9  =  Vll,  for  the  second; 
which  equations  give, 

tc  =  81,    and    y  =  72. 

Page    156. 

(3.) 
Let  aj,  y,  and  z  denote  the  separate  ages  of -4,  J?,  and  C. 
Then,  cc  =  2y,  y  —  Sz, 

and,  X  +  y  '\'  z  =  140  ; 

fi'om  which  we  find, 

03  =  84,    y  =  42,    and    z  =  14. 

Or,  this  example  may  be  solved  ^vith  a  single  unknown 
quantity.    Thus:  let    x  =   C^s  age;  then, 

(7'5  age  =z    a?, 

^'5  age  =  3cc, 

A^s  age  =  Ccc, 
and,  X  +  dx  -\-  6x  =  lOaj  =  140; 

whence,  a;  =  14. 


156. J  EQUATIONS  OF  THE  FIRST  DEGREE.  ^9 


(4.) 

Let  X  denote  the  cost  of  the  horse ;  y,  the  cost  of  the 
harness ;  and  z^  the  cost  of  the  chaise. 
Then,  x  -\'  y  ■\-  z  —  £60 ; 

also,  X  =  2y,    and  z  =  2(cc  +y)  ; 

from  which  we  find  the  several  answers. 

But  we  may  resolve  the  question  by  means  of  but  a  single 
unknown  quantity.  Thus,  let  x  denote  the  price  of  the 
harness. 

Then,  x  =  the  cost  of  the  harness, 

2x  =  the  price  of  the  horse, 
ilnd,  Qx  =  the  cost  of  the  horse  and  harness ; 

Also,    X  +  2x  +  Qx  =  9x  =  £60 ;    or,    cc  =  £6  135.  4:d. 

(5.) 
Let  0?,  y,  and  z,  denote,  respectively,  the  three  parts. 
Then,  x  +  y  ~\-  z  =  36  ; 

also,  -X  =  -V,    and    -y  =  -z: 

'  2  3*^'  3^         4   ' 

or,  3a;  ==  2y,    and    4y  =z  3z; 

from  which  we  find,  tc  =  8,  y  =  12,  and  s  =  16. 

This  example  may  be  resolved  by  only  two  unknown 
quantities.  Thus,  let  x  and  y  represent  the  first  and  second 
numbers,  then  will  36  —  cc  —  y,  denote  the  thuxl,  and  we 
shall  have, 

-X  =  -y,     or    3a;  =  2y; 

and  -y  =  ■ ^ ,    or    4y  =  108  —  3x—  Sy; 

from  which  we  find,  x  and  y,  as  before,  equal  to  8  and  12. 


30  KEY  TO  DAVlKd'  JSliW  KL15MENTAEY  ALGEBRA.        [156. 

(6.) 

Let  X,  y,  and  a?,  denote,  respectively,  the  fractions  of  the 
work  which  A^  J5,  and  C  would  do  in  a  single  day,  and  let 
the  whole  work  to  be  done  be  denoted  by  s.  Then,  in  one 
day,  A  would  do  xs  of  the  work ;  in  two  days,  2xs  of  the 
work ;  in  three  days,  dxs  of  the  work,  &c. ;  and  the  same 
for  the  others.     Hence,  by  the  conditions, 

8XS  +      8^/5    r=    5, 
9xs  +      9^5    =    5, 

10y5  +  lOys  =  s; 
and,  by  dividing  by  s,  we  have, 

8x  +    8y  =  1, 

9a;  +    9z  =  1, 

lOy  +  10^  =  1  ; 

from  which  equations  we  find  x  =  j^-^-^^  y  —  ^4J_,  and  z  —  -^^-L., 
the  parts  of  the  work  that  each  person  will  do  in  a  single 
day. 

Then,  if  each  can  do  in  one  day  the  part  of  the  work 
represented  by  each  of  these  fractions,  it  is  plain  that  the 
number  of  times  which  1  contains  each  fraction,  will  express 
the  number  of  days  in  which  each  person  would  do  the 
whole  work  ;  that  is, 

A  would  do  it  in       —   =  — — -  —  1*4  —    days; 
49  49  49         "^    ' 

^720 

7?    •  1  V20  .23     , 

•^    '"^  41    =  IT   =   ^^41     '^^^'' 

720 

n    '  1  '^20  ^^7       , 

<7   ^  ^  =  IT  =  23--    days. 


157.]  EQUATIONS  OF  TIU:  IIKST  DEGREE.  31 

Let  cc,  2/,  and  ^,  denote,  respectively,  the  sums  mth  whicli 
each  began  to  play. 

Then,  x  -{:  y  +  z  =  600 ; 

first  game,  a;  +  -  y  =  2y, 

second  game,  x  +  -7/  —  z  =  x; 

from  which  we  find,  x  =  $300,  y  =  |200,  and  z  =  $100. 

(80 
Let  Xj  y,  and  ^,  denote,  respectively,  the  sum  of  each. 
Then,  x  +  y  +  z  ==   3640  ; 

second  condition,         x  +  400  —  y  —  400  +  320, 
third  condition,  2/  +  140  =  2  —  140; 

from  which  we  have,  x  =  800,  y  =  1280,  and  z  —  1560. 


(9.) 

Let  X  denote  the  amount  of  the  bill  in  dollars,  y  the 
amount  possessed  by  A^  and  z  the  amount  possessed  by  B ; 
and  let  it  be  remembered  that  C  has  |8. 

Then,  first  condition,         2/  +  7^  =  ^  5  ' 

second  condition,  z  +  1   =  x; 

third  condition,  -  +  8  =  x: 

'  2 

from  which  we  have,  x  =  $13,  y  =  $10,  and  z  =  $12. 


32  KEY  TO  DAVIES'  NEW  ELEMENTAET  ALGEBEA.        [157. 

{  10.) 

We  may  again  remark  here,  that  if  the  rate  of  interest  be 
divided  by  100,  and  the  quotient  multiplied  by  the  principal, 
the  product  will  always  be  the  amount  of  interest. 

Let  X  denote  the  rate  of  interest  received  on  the  1st  sum ; 
then,  aj  +  1  and  cc  +  2,  will  be  the  other  rates.  Let  y  de- 
note the  capital  of  the  first ;  then,  y  +  10000  and  y  -\- 15000, 
will  denote,  respectively,  the  capital  of  the  second  and  third. 

Then,  by  1st  condition, 

and,  by  2d  condition, 

-^  =  ^ili  X  (2/  +  15000)  -  1500; 
100  100  ^^  ^  ^  ' 

that  is,  xy  =  xy  -\-  lOOOOcc  +    2/  —  TOOOO, 

and  xy  z=z  xy  +  15000a;  +  2y  —  120000; 

or,  0  =  10000a;  +    y  —  70000, 

and  0  =  15000a;  +  2y  -  120000; 

from  which  we  find,  a;  =  4,  and  y  —  30000 ;  and  hence  the 
other  two  sums  are  easily  found. 

(11.) 

Let  X  denote  a  daughter's  share. 
Then,  2a;  =  what  each  son  received  ; 

also,  3a;  +  4a;  =  what  the  children  received ; 

3a;  +  4a;  +  1 000  =  widow's  share ; 
hence,        6a;  +  8a;  +  1000  —  15000; 
or,  14a;  =  15000  —  1000  =  14000; 

or,  X  =  1000. 


158.] 


EQUATIONS  OF  THE  FIRST  DEGEEE. 


33 


12.) 
Let  the  sum  to  be  divided  be  denoted  by  x, 

X 


Then, 


A's 

share  =  -  ~  3000, 
2 

B's 

X 

share  =  -  —  1000, 
3 

G's 

X 

share  =  t  +  800 ; 

X    =z 

l  +  f  +  f-3200; 

then  the  sum, 

or,  24a;  =  12aj  +  Sx  +  6x  —   V6800; 

hence,        2x  =   ^76800;  or  x  =   $38400. 

(13.) 

Let  the  values  of  the  horses  be  denoted,  respectively,  by 
X,  y,  and  z. 

Then,  jc  +  220  =  2/  +  ^ ; 

2d  condition,  y  +  220  =  2{x  +  z)  =  2x  +  2z ; 
3d  condition,  z  +  220  =  S{x  +  y)  =  3aj  +  3y ; 
from  which  we  find,  x  =  20,  y  r=:  .100,  and  z  =  140. 

(14.) 

Let  the  number  of  guns  be  denoted  by  cc,  the  number  of 
soldiers  by  y,  and  the  sailors  by  z.  Now,  as  there  are  22 
seamen  to  every  three  guns,  there  will  be  ^/-  seamen  to  each 
gun,  and 


22 


X  cc,  for  X  guns;  hence. 


1st  condition  gives. 


22 
z  =z  —  X  x+  10; 


or, 

2d  condition, 
2* 


Sz  =  22a;  +  30    .     .     .     .  (1.) 

y  +  ^  =  5(y  +  a;)   =  5y  +  5x.      (2.) 


34  KEY  TO  DAVIES'  NEW  ELEMENTAKY  ALGEBRA.       [158 

Now,  if  we  denote  the  number  of  slain  by  5, 
then,  2/  +  ^  —  5  =  the  survivors,  and 

l{y  +  ^-  s)  =z  s,    or    s  r=  -(y  +  s) ; 

1  4 

then,       y  ^  z  —  -(y  +  z)  =  ~{y  +  z)  =  survivors,  and 
o  o 

4  13 
by  3d  condition,         -(y  +  g)  4-  5  ==  ^x ; 

5  2 

that  is,  8y  +  82  +  50  =  65a; ;      ...     (3.) 

from  which  three  equations  we  find,  a;  =  90,  y  =  55,  and 
z  ■=:  670. 

(15.) 

Let  x^  2/,  and  ^,  denote  the  sums  which  they  respectively 
had  at  first. 

Then,  x  —  y  —  z  —  what  A  had, 

2y  =  what  JB  had, 
and  2z  =  what  (7  had, 

after  the  division  with  A,    Also, 

2x  —  2y  —  2z  =  what  A  had, 
2y  •—  {x  —  y  —  z)  —  2^  =  what  JB  had, 
and  Az  =  what  (7  had, 

after  the  division  with  JB.    Again, 

4x  —  4y  —  4z  =  what  A  had, 
4y  —  2{x  —  y  ^  z)  —  4z  =  what  JB  had, 
4s  —  (2a;  —  2y  —  2;^)  —  [2y  —  (a;  —  y  —  s)  —  2^]  = 
what  C  had,  after  his  division  with  A  and  i?. 

But  these  three  last  sums  are  all  equal  to  each  other,  and 


159.]  EQUATIONS  OF  THE  FIRST  DEGREE.  35 

the  sum  of  a?,  y,  and  s,  is  equal  to  96.     Hence,  after  reduc- 
ing, we  have, 

a;  +  2/  +  ^  =  96, 

Gx  —  lOy  =  2z, 

and  5x  —    3y  =  llz; 

from  which  we  find,  x  =  $52,  y  =  $28,  and  s  =  $10. 

(160 

Let  the  parts  be  denoted  by  cc,  y,  and  z. 
Then,  cc  +  y  +  ^  =  a, 

and,  X  :  y  :  :  7n  :  n;    or,    ^cc  =  my ; 

also,  y  :  z  ::  p   :  q;    or,    qy  =  pz; 

From  these  equations  we  find, 

_         amp  _  anp  __  anq 

'"  mp-^-np+nq^       ~~  mp-\-np-\rnq'^       ~~  mp-\-np-\-nq 

( IV.) 

Instead  of  denoting  the  parts  of  the  work  done  by  -4,  B, 
and  C,  by  jc,  y,  and  z,  as  in  Example  6,  page  156,  let  us 
denote  the  times  in  which  each  would  perform  the  work, 
respectively,  by  x,  y,  and  z,  and  denote  the  work  to  be 
done,  by  1.     Now,  if  it  takes  A,  x  days  to  do  the  work,  he 

will,  in  one  day,  do  a  part  of  the  work  denoted  by  - ;  hence, 

X 

-  =  the  part  A  can  do  in  a  day, 

X 

-  =  what  B  could  do  in  a  day, 

y 

-  =  what  G  could  do  in  a  day ; 
z 


36  KEY  TO  DAVIES'  NEW  ELEMENTAET  ALGEBRA.       [150. 

and  these,  multiplied  by  any  number  of  days,  would  give 
what  each  could  do  in  those  days.     Hence, 

12        12  111 

1st  condition, 1 =  1 ;    or,    -  +  -=--  :     (1.) 

20    ,    20         ,  11  1        ,^x 

2^'  7+T  =  ^5    or,    -  +  -^=2-5'    (2-) 

3d,  l-^ii^l;    or,    i  +  i  =   i.     (3.) 

X  Z  ^  ^     X  Z  lb  ^      ^ 

Subtracting  the  second  equation  from  the  first,  we  have, 

l_l___l._JL-_^_i. 
X       z   '^   12        20   ~"  240   ""  30* 

Then,  adding  the  third  to  this,  we  have, 

2  1,1  3  60     '^ 

^  =   30  +  15  =  50'    "^'    ^  ==¥   =   '"' 

and  y  =  30,    and    z  =  60. 

Now,  the  three  together  could  do  in  one  day, 

20  "*"  30  "^  60   "   60        60        60   ~  60   ~   10 ' 

of  the  work ;  hence,  they  could  do  the  whole  work  in  ten 
days. 

Page    204. 

(6.) 
Let  the  number  be  denoted  by  x. 

*Aien,  ~x  X  ^  =  108; 

that  is,  j-x^  =  108;    or,    cc^  =  1296; 

hence,  x  =   ^1296  =  36. 


204.]  EQUATIONS  OF  THE  SECOND  DEGREE.  37 

(7.) 

Let  the  number  be  denoted  by  x. 

Then,  ~x  x  -x  -^  10  =z  3; 

5  6 

x"^  x^ 

that  is,  —---10  =  3;    or,    — —  =  3  ; 

'  30  '        '    300  ' 

hence,         x^  —  900,    and    x  ~   -^^900  =  30. 

(8.) 

Let  the  number  be  denoted  by  x. 

x^ 
Then,  aj^  +  18  =  -+  30^; 

hence,  2cc  +  36  =  cc^  +  61 ; 

consequently,  x  —   -y^  =  5 

(11.) 

Let  X  denote  the  greater  number ;  then,   -a;  will  denote 
the  less. 

Then,  x^  -  {^x\''  =  28; 

9 

that  is,  a;2 cc^  =  28  ; 

16  ' 

or,  16x^  —  9x^  =  448,    and    ^x'^  =  448; 

hence,  x^  =  64,    and    x  =  S. 

(12.) 
Let  x  denote  the  greater  number ;  then  —x  will  denote 
the  less. 


38  KEY  TO  DAVIES'  NEW  ELEMENTAKY  ALGEBRA.        [208 

25 
Then,  aj2  ^        ^,2  ^^  534^ 

and,  121cc2  +  25aj2  =  70664; 

hence,  146aj2  =::=  70664,    and    x^  =484; 

hence,  a;  =  22. 


Denote  the  age  of  the  elder  by  x ;  then   -    will  denote 


(13.) 

3  e 
the  age  of  the  younger. 

Then,  c«2  -  ^^2  _  240, 

and,  16aj2  —  cc^  =  3840;    or,    x  =  16. 

Page    208. 

(5.) 
Let  the  two  numbers  be  denoted  by  x  and  y. 
Then,  by  first  condition, 

cc2  +  2/2  =  117; 
by  second  condition, 

aj2  _  2/2  =:     45  ; 
from  which  we  have, 

a!  =  9,    and    9/  =  6. 

(6.) 

Let  the  two  numbers  be  denoted  by  x  and  y. 
Then,  aj2  +  2/2  -_.  ^^ 

and,  aj2  —  2/^  =  ^5 


hence. 


'»       ^  ="   V  ^2~'    ^^*    ^  ^  V~ 


208.]  EQUATIONS  OF  THE  SECOND  DEGREE.  39 

(9'.) 

Denote  tlie  numbers  by  x  and  y. 
Then,  a;  :  2/  :  :  3  :  4 ;    or,    4a;  =  3y, 

and,  cc2  j^  y^  —  225 ; 

from  which  we  have, 

03  =  9,    and    y  —  12. 

(8.) 

Denote  the  numbers  by  x  and  y. 
Then,  x  \  y  \\  m  \  n\    or,    nx  =z  my^ 

and,  «;2  +  2/2  _  ^2^ 

Squaring  the  first  equation,  and  multiplying  the  second 
by  w?^  we  have,  by  transposing  in  the  second, 

n^x^  =  rn?y'^^ 

and,  n%^^  =  w?o?  —  m?y'^ ; 

and  then,  by  addition,  we  obtain, 

x^(m?  +  'n?)  =  nh^a\ 

T                        ma  -  na 

and,       X  —  -    _;    also,    y 


(9.) 

Let  the  largest  number  be  denoted  by  2cc ;  then  the  less 
will  be  denoted  by  x.    We  shall  then  have, 

4i»2  —  a;2  =  75, 

and,  3i«2  =  75,    and    a;^  =  25 ; 

or,  a;  =  5. 


40  KEY  TO  DAVIES'  NEW  ELEMENTAET  ALGEBRA.        [i^08. 

(10.) 

Let  the  numbers  be  denoted  by  x  and  y. 
Then  x  \  y  ::  m  :  n\    or,    nx  =  my^ 

and,  x^  ~  y"^  =z  b^; 

from  which  we  find, 

mb  T  nb 

:,    and    y  = 


(11.) 

Let  the  amount  in  dollars,  placed  at  interest,  be  denoted 
by  X. 

Then,         — —  x  cc  =  the  interest  for  one  year, 
'100 

4 
and,  — -  X  cc  =  the  interest  for  six  months, 

^"^^^^  Too  ^  ^  ^   25   "^   562500, 

and,  x^  =  14062500,^   and    x  =  3750. 

(12.) 

Let  x  denote  the  number  of  women,  and  y  the  number 
of  boys. 

Then,  x  :  y  :  :  S  :  4;    or,    4cc  =  3y, 

and,  X  =  -y; 

and,  X  +  y  =  the  number  of  persons, 

^  =  what  the  boys  receive. 

and,  2y  =  what  the  women  receive. 


221.]  EQUATIONS  OF  THE  SECOND  DEGREE.  41 

Then,  ^  t  ^  +  2y  r=  138, 

and,  aj  +  2/  +  4y  =  276, 

Then,  _  2/,  +  2/  +  4y  =  276, 

and,  3y  +  4y  +  16y  =  1104; 

or,  23y  =  1104,    and    y  =  48. 

Note. — This,  it  will  be  seen,  is  an  equation  of  the  Jirst  degree^  and  is 
placed  among  those  of  the  second  degree,  to  lead  the  student  to  have 
confidence  in  his  own  methods,  and  not  to  rely  too  implicitly  on  the 
arrangements  of  the  author. 


Page  221.) 
( 10.) 

Clearing  of  fractions, 

a^n^  +  b'^91^  —  2bn'^x  +  n^x'^    =:  m'^x'^. 
Transposing,  arranging  the  terms,  and  factoring,  we  have, 
[n^  —  m^)x'^  —  2bn^x  =   —  ahi^  ~  b^-ri^ ; 
dividing  by  the  coefficient  of  cc^,  we  have, 


y.2 


25^2  -—  a^nP'  —  y^n? 


'X   = 


w^  —  m^  n^  —  m 


2  


Completing  the  square, 

2bn^  lh%^  -d'n'^—'b'^n^  .         Ihi"^ 

-,x  + 


^2,2  _  ^2*^  '^   (,2,2  _  ^^2)2  ^^2   _^2  ^  (^^2  _  ,^^2)2' 

Multiplying  the  numerator  and  denominator  of  the  first 


42  KEY  TO  DAVIES'  NEW  ELEMENTAPwY  ALGEBRA.        [2'f3. 

term  of  the  second  member,  by  n^  —  m^,  so  as  to  reduce 
both  to  a  common  denominator,  we  have, 

"""  (n^  —  rn^Y 

Cancelling  like  terms,  and  factoring, 

then  extracting  the  root  of  both  members, 


03  — .    _-     -J-    — I . 

Transposing  and  factoring, 


n' 

n 

rti^ 

n^ 

n 

m^ 

n^ 

— 

w? 

a;"  =  -^ ^  {bn  —  -y/a^m^  +  ^^^2  __  ^2^aj^ 


Page  243, 

(4.) 

Let  X  denote  the  greater  number ;  then   11  —  a  will 
denote  the  less. 

By  the  conditions  of  the  question, 

x^  +  121  —  22a;  -^  x^  z=z  61 ; 
whence,  1x^  —  22a;  =   —  60  ; 

or,  x^  —  11a;  =   —  30  ; 

whence,  a;'  =  6,     and    a;"  =  5. 

Either  root  will  satisfy  the  conditions  of  the  problem. 


243.J  EQUATIONS  OF  THE  SECOND  DEGREE.  4:3 

(5.) 

Let  X  denote  the  greater  number ;  then,  since  the  differ- 
ence of  the  two  numbers  is  3,  cc  —  3  will  denote  the  less. 

By  the  conditions  of  the  question, 

aj2  +  cc2  —  6cc  +  9  =  89 ; 
hence,  203^  —  6a?  =  80, 

and  ic^  —  3a3  =  40; 

hence,  cc'  =   +8,     and    cc"  zz:   —  5. 

The  plus  value  belongs  to  the  problem ;  hence,  the  num- 
bers are  8  and  5. 

(6.) 

Let  X  denote  the  number  of  sheep  which  he  purchased. 

Then,  —     =  the  cost  of  a  single  sheep  ; 

or,  z=:  the  cost  of  a  single  sheep  in  shillings, 

,               1200  X  15         18000  ,    n..,         . 

and =  =  cost  of  fifteen  sheep. 

XX'  ^ 

Then,  the  number  of  sheep  sold  will  be  represented  by 
a;  —  15,  and  2(03  —  15)  =  the  amount  of  profit.  Now, 
had  there  been  no  profit,  the  amount  received  for  the  sheep 
would  have  been  just  equal  to  the  cost,  less  the  value  of  the 
fifteen  unsold  ;  and,  consequently,  the  amount  received,  less 
the  profit,  must  be  just  equal  to  this  difference.  That  is, 
reducing  to  shillings, 

1080   -   2{x  ~  15)   =   1200  -  H^; 

X 

hence,       1080aj  —  2x'^  +  SOx  z=  1200i«  —  18000; 

and,  by  reducing  and  dividing  by  the  coeflicient  of  x\  we 
have, 

j»2  +  45a;  =  9000 ; 

from  which,  by  taking  the  positive  value,  we  have,  x  =  75. 


44  KEY  TO  DAVIES'  NEW  ELEMENTAEY  ALGEBEA.       [243. 

{'if') 

Let  the  number  of  pieces  be  denoted  by  x ;  and  by  re- 
ducing the  cost  and  the  amount  received  to  shillings,  we 
find  that  he  paid  675  shillings,  and  sold  for  48  shillings  per 
piece, 

675 

Then,  =  the  price  ]3er  piece  in  shillings, 

and  ASx  =  what  he  received  for  the  whole. 

But  what  he  received,  minus  what  he  gave,  must  be  equal 
to  his  -profits ;  that  is,  to  the  cost  of  a  single  piece.     That  is, 

675 

48a;  -  675  =  , 

X 

and  48a;2  -  675aj  =  675  ; 

675  675 

-^'"--4^^  =   I8-' 
and  by  completing  the  square, 

,       675      .    (675)2         675    ,    (675)^ 
48      ^    (96)2    -    48    ^    (96)2 
_   1350        (675)2 
~     96     "^    (96)2 
_   1350  X  96        (675)2 
~         (96)2        "^      96 
585225 


-      (96) 

Then,  by  extracting  the  square  root  of  both  members,  and 
taking  the  positive  root,  which  answers  to  the  question  in 
its  arithmetical  sense,  we  have, 

__  675  _  765^ 

^  ""    96    ~    96  ' 

675    ,    765         1440 

and*  X  =  —  H =  =  15, 

'  96  96  96 


243.]  EQUATIONS  OF  THE  SECOND  DEGREE.  45 

(8.) 

Let  X  denote  the  greater  number ;  then,  since  the  differ- 
ence of  the  two  numbers  is  equal  to  9,  the  less  will  be  de- 
noted by  cc  —  9  ;  and  their  sum,  by  2cc  —  9.  By  the  con- 
ditions of  the  question, 

(x  —  9)x  =  266  ;     or,     x^  —  9x  =  266  ; 

from  which  we  find,   cc  =  14 ;   and,  consequently,  the  less 
number  is  equal  to  5. 

(9.) 

Let  the  number  be  denoted  by  x. 
Then,  (10  —  x)x  =z  21 ; 

hence,  lOcc  —  x-  =  21 ; 

or,  x^  ■—  10a;  =   —  21. 

Completing  the  square, 

i«2  -  lOa;  +  25  r;=:    -  21  +  25  =  4 ; 
hence,  xz=z5±^=:5±:2  =  7^   and  3. 


(10.) 

Let  X  denote  the  distance  traveled  in  one  hour;  then, 
X  —  2  will  denote  the  distance  when  he  travels  2  miles  an 
hour  slower. 

105 
Then,     —  =  the  number  of  hours,  under  the  first  sup- 
position ; 

J.05 

and, =  the  number  of  hours,  under  the  second 

X  —  2  ' 

supposition. 


46  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.       1 244 


But,  by  tlie  conditions, 

105          105 
a;  -  2         X     ~      ' 

whence,         105a;  —  105aj  +  210  —  Qx^  — 

12a;; 

that  is,                            6a;2  —  I2x  =  210, 

a;2  -    2x  =:     35. 

• 

X   =    1 


(11.) 

Let  X  denote  the  number  of  sheep ;  then, 

—  =  what  was  paid  for  each ;  and, 

448 

[-  2  =  what  he  would  have  paid,  under  the 

X 

second  supposition, 
llien,  by  the  conditions  of  the  question, 

—  +  4  =  x; 

whence,  896  +  4a;  =  a;^; 

from  which  we  find,  a;  =  32. 

(12.) 

Let  x  denote  the  greater  number.  Then,  since  the  differ- 
ence of  the  numbers  is  7,  x  —  1  Avill  denote  the  lesser, 
and  their  sum  will  be  denoted  by  2a;  —  7.  By  the  condi- 
tions of  the  question, 

(2a;  —  1)x  =  130;    or    2a;2  —  1x  =  130; 

whence,  we  find  the  plus  value  of  a;,  which  belongs  to  the 
problem,  to  be  10;  hence,  the  numbers  are  10  and  7. 


244,]  EQUATIONS  OF  THE  SECOND  DEGKEE.  47 

(13.) 

Let  X  denote  the  greater  number ;  then,  since  their  sum 
is  100,  100  —  a;  will  denote  the  less.  By  the  conditions 
of  the  question, 

a;2  +  10000  -  200a3  +  x^  =  5392; 

hence,      2aj2  —  200a;  =  5392   -  10000  —   —  4608; 

or,  x^  —  lOOcc  =   —  2304  ; 

whence,  we  find  the  positive  value  of  cc,  which  belongs  to 
the  problem,  to  be  64. 

Let  X  denote  the  number  of  feet  in  the  side  of  the  smaller 
court  yard;  then,  x  ■\-  \1  will  denote  the  side  of  the 
larger,  in  feet.  Then,  since  each  stone  contains  1  square 
foot, 

(x  +  12)  ~  cc2+  24aj  +  144  =  number  of  stones  in  larger, 

and,  Qi?                         z=  number  in  smaller ; 

hence,  2cc2+ 24cc  +  144  =  2120; 

or,  x'^-h  12x  z=z  1060  -  12  -  988 ; 

whence,  x  =  26. 

(15.) 

Let  X  denote  the  number  of  persons. 

240 
Then,     =  what  each  received  in  the  first  distribution ; 

X 

and,    =  what  each  received  in  the  2d  distribution. 

X'\-  4 


48  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.        [244. 

By  the  conditions  of  the  question, 

240          240 
— -  =  10; 

X  X    -[-    4: 

whence,      240a;  +  960  —  240a;  =  lOx^  +  40a;; 
whence,  lOa;^  +  40a;  =  960  ; 

or,  x^  -\-    4:X  =     96 ; 

whence,  cc'  =  8,    and    a;"  =  —  14. 

The  plus  value  alone  belongs  to  the  problem ;  and  the 
numbers  of  persons  were  8  and  12. 

(10.) 

Let  X  denote  the  number  of  dollars  in  A's  capital.  Then, 
since  he  received  520  dollars  as  capital  and  profits,  he 
gained,  in  dollars,  520  —  a;,  in  12  months;  hence,  the 
number  of  dollars  which  he  gained  in  1  month,  will  be 

denoted  by   — • 

^  12 

Now,  B^s  gain  was  the  whole  gain  diminished  by  A^s ; 

hence,  B''s  gain  is  denoted  by   360  —  (520  —a;)  =  a;—  160; 

since  B'^s  capital  was  in  trade  16  months,  his  gain  in  1  month 

will  be  denoted  by 

^         16 

But  the  gains  of  each  in  1  month  must  be  proportional  to 
the  capitals  they  employed ;  hence, 

520  —  a;     x  —  160 


X  :  600 


12        *        16       ' 


X  —  160         ^^^        520  —  X 
or,  X  X  — -^ —   =  600  X  j^—  • 

Clearing  of  fractions,  and  reducing, 

a;2  —  160a;  =  416000  —  800a;; 
or,  x^  +  640a;  =:  416000; 

whence,  we  find  the  plus  value  of    a;  =  400. 


255.]  EQUATIONS  OF  THE  SECOND  DEGREE.  49  . 

*  Page    255. 

(3.) 
Let  X  denote  one  number,  and  y  the  other. 
Then,  by  the  conditions, 

aj  +  y    =     8 (1.) 

and,  x^+  y^  =  S4:    .    .    \    .    .    (2.) 

From  the  first  equation,  we  have, 

X  =z  S  —  y,    and   cc^  -_  54  _  jgy  +  y^. 
Substituting  for  aj^,  in  Equation  (  2 ),  we  have, 

64  _  162/  +  2/^+  2/^  =  34; 
whence,  2y^  —1^2/  =   —  30, 

2/2  -    82/  =:   -  15. 

Hence,  2/'  =  ^?    ^^^    2/"  =  ^• 

Hence,  cc'  =  3,    and   a;"  =  5. 

(4.) 

Let  X  denote  the  first  number,  and  y  the  second. 
Then,  by  the  conditions, 

X  :  y  ::  y  :  IQ; 

whence,  16x  =  y^ (1.) 

cc2+  2/2  =  225 (2.) 

Substituting  the  value  of  2/%  from  Equation  ( 1 ),  we  have, 

x^  +  Ux  =  225; 
whence,  a?  =  9,    and    y  z=  12, 


L 


3 


50  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.      [256. 

(5.) 
Let  X  denote  the  less,  and  y  tlie  greater. 
Then,  by  the  conditions, 

2/  ::  3  ;  5; 

whence,  bx  =  Zy  ,     .     .    .     .     .     (1.) 

2d  condition,  x^  ■\- y'^  =  1666 (2.) 

From  Equation  (1),  we  have, 

25 
Substituting  this  value  in  Equation  (  2  ), 

-~  +  y^=-  1666  ; 

whence,  342/^  =  41650; 

and  2/2  =  1225  ;    or,    y  =  35. 

Hence,  the  numbers  are  21  and  35. 

(6.) 

Let  X  denote  the  less  number,  and  y  the  greater. 
Then,  by  the  conditions, 

y  —  X  =  7 ;    whence,    y  =  jc  +  7, 

and,  ^  +  30  =  x\ 

Substituting  for  y,  its  value,  we  have, 

^^^  ^  "^^  +  30  =  ic2;     or,     ic^  _^  7aj  +  60  =  17?\ 
whence,  x^  —  ^x  —  60; 

or,  X  —  12,     and    y  —  19. 


256.]  EQUATIONS  OF  THE  SECOND  DEGKEE.  51 

Let  the  numbers  be  denoted  by  x  and  y, 

X  +  y   =      5 (1.) 

aj3  +  2/3  =     35 (2.) 

x^  +  Zx^y  +  3aj?/2  -\-  y^  =  125,   by  cubing  (1), 

Zx^y  +  3CCI/2  —     90,   by  subtraction, 
Zxy(x  -\-  y)    =     90,   by  factoring, 
15xy    —     90,   fromEq.  (1); 

hence,  ^2/   =       6 (3.) 

Combining  (1)  and  (3),  we  find, 

a;  —  2,     and    y  =  3. 

(8.) 

Let  X  denote  the  greater,  and  y  the  less. 
Then,  by  the  conditions, 

cc  +  y  :  cc  :  :  11  :  7; 
whence,  1x  +  1y  —  lice; 

or,  aJ  =  ^2/;    or,    aj^  ^  _y2. 

2d  condition,  a;^  __  ^/^  _     132. 

whence,  by  substitution, 

-^-y'^=     132; 

or,  491/2  -  162/2  =  2112, 

332/2  —  2112, 

2/2  =       64 ; 

and,  2/   =   ^/^  =  ^> 

and  X  is  found  to  be  equal  to  14. 


52  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.       [256. 

(9.) 

Let  X  and  y  denote  the  numbers. 
Then,  by  the  conditions, 

a?  +  2/  =  100 (1.) 

ajy  :  cc2  +  2/^  :  :  6  :  13  ; 
hence,  13ajy  =  ea:^  +  6^/^    .     .     .     .     (2.) 

From  Equation  (  1 ),  we  have, 

cc  =  100  ~  2/,     and     x^  =   10000  —  200y  +  y^. 

Substituting  these  values  of  x  and  aj^,  in  Equation  (  2 ),  we 
shall  have, 

13y(100  —  y)  —  6(10000  —  200y  +  y'^)  +  6y2. 
Perfoi:ming  the  operations  indicated,  we  have, 

1300y  —  13y2  —  60000  -  1200?/  +  6?/2  +  e*/'-^ ; 
or,  25^/2  _  2500y  —   —  60000 ; 

or,  2/^  —    lOOy  =   —    2400 ; 

whence,  y  =  60,     and    a;  =  40. 

(10.) 

Let  X  denote  the  distance,  in  miles,  traveled  by  -4,  and 
y,  the  distance  traveled  by  JB, 

Then,  by  the  conditions, 
1st  condition,  cc  —  2/  =  18 (1.) 

Now,  the  rate  of  travel,  or  the  distance  traveled  in  a 
single  day,  will  be  found  by  dividing  the  distance  by  the 
number  of  days ;  hence, 

-^  =  what  A  traveled  in  1  day ; 
15|- 

and,  —  =  what  JB  traveled  in  1  day. 

28 


256.]  EQUATIONS  OF  THE  SKCOND  DEGREE.  53 

[N'ow,  the  e^itire  distance  traveled  by  A^  divided  by  what 
he  traveled  in  1  day,  will  give  the  time  he  Avas  traveling ; 
and  the  same  for  J?.     But  these  times  are  equal ;  hence, 

^        __       y 

(y\  ~  (-)' 

\15}/  \28/ 

and,  by  reducing, 

15faj2  _      282/2; 

or,  reducing  to  4ths, 

63aj2   ~    112?/2. 

Substituting  the  value  of  x^^  from  Equation  (1),  we  have, 

63(324  +  36?/  +  y^)  =^   lUy^ ', 

or,  20412  +  22682/  -f  63y2  =   U2y^; 

492/2  —  22682/  =  20412  ; 

2        2268      __   20412 

^        igT^  ~"     49    ' 

Completing  the  square. 


2268  (2268)2  20412    ,    (2268)2 

^  49  *^  (98)2     -       49      ^     (98)2    ' 


2        ^^^^       I    (2268)2   _   40824  X  98        (2268)^^ 
and       y  -  -—y  +  -j^^  -  — (^p        +  798)^  ' 

which,   after  performing  the   operations  indicated,  gives, 
2/  =  54,   and,  consequently,   x  =  72, 

(11.) 

Denote  the  less  number  by  x.    Then  the  greater  will  be 
denoted  by  cc  +  15  ;  and  we  shall  have, 

(^  +  15)a?  ^ 

2  ' 

and,  dividing  by  a?,  a;  +  15  =  2a;2; 


54  KEY  TO  DAVIKS'  NEW  ELEMENTARY  ALGEBRA.       [256. 

and,  by  transposing,        x'^  —  -x  =  7.5  ; 

and,  completing  the  square, 

x^  —  .5x  +  .0625   =r  7.5625  ; 
hence,  aj  :=   .25  ±  V^5625   =  3, 

by  taking  the  positive  root. 

(12.) 

Let  the  greater  number  be  denoted  by  x,   and  the  less 
^7  2/- 

Then,  by  first  condition,  (x  +  y)^  =  77  ....  (1.) 

by  second  condition,  (x  —  y)y  ==  12  ....  (2.) 

that  is,  aj^  -f  ccy  r=  77  .  .  .  .  (3.) 

and,  ccy  —  2/^  =:  12  .  .  .  .  (4.) 

By  adding,  we  have,  x^  —  y"^  +  2aji/  =  89  ; 

and,  by  transposing,  x^  —  y'^  =  89  —  2xy, 

K  we  multiply  the  first  and  second  equations  together, 
we  obtain, 

{x^-  —  y'^)xy  =z  924  ; 

and  hence,  x"^  —  y"^  =  • 

''  xy 

Placing  this  value  of  cc^  —  ^/^  equal  to  that  found  above, 
and  Ave  have, 

924 

89  —  2xy  =  ; 

xy 

or,  S9xy  —  2a;y  =  924 ; 

and,  placing  xy  =  z,   we  obtain, 

89s  -  2^2  =  924 ; 


257.]  EQUATIONS  OF  THE  SECOND  DEGEEE.  65 

and  hence,  by  changing  the  signs  and  dividing,  we  have, 

a;2  __  44.5^  —   _  462. 
Then,  by  completing  the  square, 

^2  -  44.5^  +  495.0625   =  33.0625  ; 


hence,  z  =   22.25  ±  -v/53.0625  ; 

or,  z  z=z   22.25  ±  b.lb  ; 

or,  s'rr  28;     and,     z^'  —  16.5. 

Snbstitutmg  the  first  value  of  z  for  xy^  in  Equations  ( 3  ) 
and  ( 4.) ,  gives  cc  r=  7,  and  y  =  4  ;  and  substituting  the 
second  value,  s"  =  16.5,  for  xy^  in  the  same  equations,  we 
find, 

(13.) 
Let  the  numbers  be  denoted  by  x^  and  y'^. 

Then,  cc^  +  ^/^  =  100 (1.) 

and,  X  +  y  =     14 (2.) 

From  the  second  equation,  we  have,  by  transposing, 

aj  =  14  -  2/; 
and,  by  squaring,      x^  =  196  —  28y  -{-  y\ 

Substituting  this  value  in  Equation  ( 1 ),  we  have, 

196  -  2Sy  +  y^  +  y^  =  100; 
and,  by  reducing,  y^  —  liy  =   —  48. 

Completing  the  square,  we  have,  - 

y2  __  i4y  _|.  49   _    _  48  +  49  =   1, 
and,  2/=z:+7±l  =  8; 


66  KEY  TO  DA  vies'  NEW  ELEMP:NTARY  ALGEBRA.       [257. 

or,  if  we  take  the  minus  sign,  then  7/  =z  Q,  If  we  take 
2/  =  8,  we  find  x  .—  6,  and  if  we  take  y  =  6,  we  find 
X  =  8 ;  hence,  the  numbers  are  64  and  36. 

(14.) 

Let  the  numbers  be  denoted  by  x  and  y. 
Then,  x  -{-  y  =  24^ 

and  XT/  =  35(cc  —  y)   =  S5x  —  35y. 

From  the  first  equation,  we  have, 
x  —  24  —  9/, 
Substituting  this  value  in  the  second,  we  obtain, 
2/(24  -  y)  =  35(24  -  2/)  -  35^; 
that  is,  24y  —  y'^  —   840  —  351/  —  35y ; 

hence,  j/^  —  94^/  =   —  840. 

Completing  the  square, 

2/2  __  94y  j^  2209   =   1369, 
and,  y  =  47  ±  37  =  84,  and   10. 

If  we  take  the  first  root,  84,  the  value  of  x  will  be  —  CO, 
and  these  two  numbers  will  satisfy  the  two  equations  of 
condition.  But  the  enunciation  of  the  question  required 
the  number  24  to  be  divided  into  two  parts,  and  this  re- 
quired that  neither  x  nor  y  should  have  a  value  exceeding 
24  ;  hence,  we  must  take  the  second  value  of  y  =  10.  This 
gives  jc  =  14. 

(15.) 
Let  the  numbers  be  denoted  by  x  and  y. 

Then,  a;  +  2/  =  8 (1.) 

and,  a;3  +  2/^  =  152      ....     (2.) 


257.]  EQUATIONS  OF  THE  SECOND  DEGREE.  57 

By  cubing  both  members  of  Equation  ( 1 ),  we  have, 

x^  -f  3cc2y  +  3CC2/2  +  2/3  =  512     .     .     (3.) 

and,  by  subtracting  the  second  equation  from  the  third,  we 
have, 

3aj2y  +  3032/2  :^   360 ; 

and,  dividing  by  3,  ive  obtain, 

x^y  +  xy"^  =  120; 

factoring,  xy{x  +  y)  =  120  ; 

but,  since,  in  Equation  ( 1 ), 

X  +  y  =2  8, 

we  have,  8xy  =  120 ;     or,     xy  =  15. 

Combining  this  with  Equation  ( 1 ),  we  readily  find, 

cc  =:  3,     and    y  =:  5. 

(16,) 

Let  the  number  of  yards  sold  by  the  first,  be  denoted  by 
x,  and  the  number  sold  by  the  second,  by  y. 

N"ow,  if  the  whole  amount  received,  for  any  number  of 
things  solely  be  divided  by  the  number  of  things,  the  quo- 
tient will  be  the  cost  of  each  thmg.  Hence,  if  24  dollars 
be  divided  by  the  number  of  yards  of  stufi*  sold  by  the 
second,  the  quotient  will  be  the  amount  per  yard  received 
by  the  first;  and,  for  a  like  reason,  12^  divided  by  cc,  will 
be  the  amount  per  yard  received  by  the  second. 

24 
That  is,     —    =   what  the  first  received  per  yard. 

12^ 
and,  — ~  =    what  the  second  received  per  yard. 

But,  the  first  sold  x  yards,  and  the  second  y  yards ;  and, 
3* 


58  KEY  TO  DAVIES'  KEW  ELEMENTAKY  ALGEBRA.        [257. 

if  the  amount  per  yard  be  multiplied  by  the  number  of 
yards,  the  product  will  be  the  amount  received.     Hence, 

24  ,    124- 

—  X  X  -\ ~  X  2/  =  35  ; 

2/  ^  . 

and,  by  the  second  condition, 

2/  —  X  =  S;     or,     y  =  x  +  S, 

Then,  by  clearing  the  first  equation  of  fractions,  we  have, 

24a;2  +  12^2/^  =  SBxy ; 

and,  by  substituting  for  y  its  value,  a:  +  3,  we  obtain, 

24a;2  +  12i(aj2  +  6a;  +  9)   =z  35a;(a;  +  3); 

that  is, 

24cc2  +  12iaj2  4-  "jBx  +  1121  =  S5x'^  +  105a;; 

and  reducing,  l-^a;^  —  30a;  :=   —  112J; 

and,  dividing  by  1^,  we  have, 

a;2  —  20a;  =   —  75  ; 

which  gives,  a;  =  10  ±  5   =  15,     and     5  ; 

^i'om  which  we  have  the  corresponding  values  of  y  =  18, 
or  y  =1  8, 

(17.) 

Let  the  highest  rate  of  interest  be  denoted  by  y,  and  the 
lowest  by  z.  Now,  as  the  incomes  are  to  be  equal,  it  is 
plain  that  the  first  sum  put  at  interest  will  be  the  least, 
which  let  us  denote  by  x.  Then,  the  larger,  or  second  part 
will  be  denoted  by  13000  —  x.  Then,  since  the  amount  of 
interest  on  any  sum  is  equal  to  the  sum,  multiplied  by  the  " 
rate,  divided  by  100,  we  have, 

by  first  condition,         x  X  ~~  =  ^13000  —  a^)  X  -— ; 


257.]  EQUATIONS  OF  THE  SECOND  DEGREE.  69 


Z 

by  second  condition,    x  x  — —  -—  360 ;  , 

by  third  condition,       (13000  —  ^-^  =  490. 

Clearing  of  fractions,,  we  have, 

xy  —  130003  —  CCS    ....  (1.) 

xz  =2  36000 (2.) 

and,             13000y  —  xy  =  49000 (3.) 

If,  now,  we  substitute  the  value  of  xz  from  Equation  (2), 
in  Equation  (1 ),  we  shall  have, 

xy  =z   13000s  -  36000  ....  (4.) 

then,  adding  together  Equations  (3)  and  (4),  we  have, 

13000y  =:  13000s  +  13000; 

or,  ISy  =   13s  +  13; 

or,  y  =  z  +  1 .     (5.) 

l^ow,  to  elimmate  x  from  Equations  (2)  and  (3),  mul- 
tiply the  first  by  y,  and  the  second  by  s,  and  we  have, 

xyz  —  36000y, 

and,       13000ys  —  xyz  =   49000s, 

and  by  adding,     13000ys  =  36000y  +  49000s; 

or,  13ys  =  36y  +  49s. 

Now,  substituting  for  y  its  value  in  Equation  ( 5 ),  we 
have, 

13s(s  +  1)  =  36(s  +  1)  +  49s; 

that  is,  13s2  +  13s  =  36s  +  36  +  49s; 

A  ■  '72  36 

and,  ,2___,  ^_. 


z^ 

72     , 

-r3^  + 

(36)^ 
(13)^ 

= 

36        /3( 
13  "^  UJ 

36   X  13 

(13)2 

z  = 

36 
13 

~  13 

= 

13 

60  KEY  TO  DAVIKS'  NEW  ELEMENTARY  ALGEBRA.       [257. 

by  completing  the  square,  we  have, 


4- 


Hence, 

The  negative  value  of  z  is  not  applicable  to  the.  ques{ion. 

(18.). 

Let  the  second  number  be  denoted  by  a;,  and  the  differ- 
ence between  the  second  and  first  by  y. 

Then,  ^  —  V  =  1st  number, 

ic  =   2d  number, 

and,  a;  +  2/  +  6  =  3d  number. 

Then,  3cc  +  6  ==  33,     and,  hence,     x  =  9; 

also,  {x  -  yY  -\-  x^  +  (x  +  y  +  6)^  =  467 ; 

that  is,  3aj2  +  120?  +  12?/  +  2y2  =  431 ; 

or,  substituting  for  cc,  its  value,  9, 

351  +  12y  +  22/2  =  431; 

hence,  y^  +  ^y  =     40, 

and,  y  =  4,     or     —  10. 

Hence,  the  numbers  are  5,  9,  and  19. 

(19.) 

Let  X  denote  the  digit  which  stands  in  the  ten's  place, 
and  y  the  digit  which  stands  in  the  unit's  place. 

Then,  lOx  -{-  y  ~  the  number. 


257.]  EQUATIONS  OF  THE  SECOND  DEGREE.  61 

By  the  first  condition, 

10^  +  y  ^  3 
xy  ' 

and,  by  the  second, 

lOcc  +  y  +  18  =  102/  +  ^5 

from  which  we  readily  find, 

£c  =  2,     and    y  =  4. 

(20.) 
Let  X  and  y  denote  the  numbers. 
Then,  x  :  y  :  i  m  :  n\     or,     y  =  —    •     •     (1.) 

and,  cc2  +  2/2  _  52 (2.) 

Substituting  the  vahie  of  y"^  from  Equation  (1 ),  we  have, 
9    .    ^^^^         J. 


or,              w?x^  + 

n^x^  - 

=  m?h 

5 

or, 

{m?  +  n^)x^  - 

= .  m% 

2            m^h 

and 

X    — 

m^/b 

and,  from  Equation  ( 1 ), 

y  = 

n^/h 

(21. 

) 

Let 

X  and  y  denote  the  numbers. 

Then, 

by  the  conditions. 

7lX 

X  :  y  :  :  m  :  n:    or    v  =  —     *     •     (!•) 


and,  cc^  —  2/^^  =  ^* 


62  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.        [258, 

Substituting  the  value  of  y^  from  Equation  ( 1 ),  we  have, 

aj2 =  5 ; 

'in^/h 


hence,  x  = 

and,  y  = 


myfb 


(22.) 

Let  the  numbers  be  denoted,  respectively,  by  a?,  y,  and  z. 

Then,  ccy  =  2,    xz  —  4,    y^  +  ^^  _.  20. 

From  the  first  equation,  we  have, 

2  2^ 

aj  =  - ;    and  hence,    —  =  4 ;    or,    ^  =  2.v. 
2/  y 

Substituting  this  value  of  s  in  the  third  condition,  we  have, 

y1  +  4i/2   :::.   20  ; 

whence,       5^/^  =20,    2/^=4,     and    y  =  2, 
Hence,  the  numbers  are  1,  2,  and  4. 

(23.) 
Let  the  numbers  be  denoted  by  cc,  y,  and  z. 
Then,  j»  +  y  +  s   =     38      ....     (1.) 

a;2+  2/2^  a:2  =  634      ....     (2.) 
and,  2/--cc  =  ^  —  2/  +  7      .     .    .     (3.) 

Adding  the  first  and  third  equations,  we  have, 

3y  =  45;     or,     y  =  15  ; 
from  which  we  easily  find, 

cc  =:  3,     and    z  =  20. 


258.]       EQUATIONS  OF  THE  SECOND  DEGREE.         63 

(24.) 
Let  the  numbers  be  denoted  by  x^  y,  and  z. 
Then,  ccy  =  a,     xz  =  5,     y^  -{-  z^  =  c. 

From  the  first  equation,  we  have,  . 

a  -,  -  az         ^ 

X  =  - .     and  hence,    —  =  o: 

y  y 

or,  az  =  5y,     and    y^  =  —z^ ; 


hence,      |V  +  ^^  ^  c;     or,     z   =  h^J —^—^, 
y  =  ^\/5^5     or,    X  =^/~r 


+  b^'        '  V  a2  4_  52 

(25.) 
Let  the  greater  number  be  denoted  by  cc,  and  the  less  by  y. 
Then,  {x  +  y)x  =  lU       .     .     .     .     ( 1.) 

{x-y)y=zU       ....     (2.) 
and  multiplying  the  equations  together,  we  obtain, 

(x2  -  y'^)xy  =  2016     ....     (3.) 
But  Equations  ( 1 )  and  ( 2 )  may  be  put  under  the  forms, 

x'^  -\-  xy  =     144, 
and,  xy  —  y^  =z  14 ;    or,    xy  =  14  +  2/2. 

and  subtracting, 

aj2  +  2/2  _  130;     or,     x'^  =  130  —  y\ 

Substituting  in  Equation  (3),  the  value  of  xy  =  144  —  a;^, 
and  then  for  x^,   its  value,    130  —  y^,   and  we  obtain, 

(130  -  2^2)  (14  _^  2/2)   =  2016; 
that  is,        1820  —  28.y2  4.  i30?/2  —  2y^  =  2016. 


64:  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.        [264. 

Hence,  y*  —  ol^/^  =   —  98. 

Then,  placing    z  =  y\   and  z^   for  y*^   we  have, 

which  gives,  by  taking  the  j)ositive  root,  which  is  the  one 
corresponding  to  the  arithmetical  enunciation, 

z^  =  49,     and,  consequently,     y^  =  49,     or     y  =  7. 

The  value  of  x  is  easily  found,  and  is  equal  to  9. 

Page  264. 

ARITHMETICAL      PROGRESSION. 

(2.) 

I  z=  a  —  {n  —  l)r. 
Make,      «  =  90,    r  =  4,     and    n  =  15 ; 
then,        I    =  90  —  (15  —  1)4  =  90  —  56  =  34. 

(3.) 

I  =  a  —  (n  —  l)r. 
Make,       a  =  100,     n  —  40,     and    r  =  2; 
then,         I    =  100  -  (40  -  1)2   =  100  —  '78  =  22. 

(4.) 

I  =  a  —  {n  —  l)n 
Make,      a  =  80,     n  =  10,     and    r  =  4; 
then,         I    =  80  -  (10  —  1)4   =   80  -  36   =  44. 

(5.) 

I  ~  a  —  {n  —  l)r. 
Make,       a  ==  600,     n  =  100,     and    r  =  5; 
then,         I    =  600  —  (100  -  1)5   =  600  —  495   =  105. 


266.]  AKmiMETICAL   PEOGEESSION.  66 

(6.) 

^  =  a  —  {n  —  l)r. 
Make,      a  =  800,     7i  =  200,     and    r  =  2; 
then,         I    =   800  —  (200  -  1)2   =   800  —  398  =  402. 

Page    266. 

(2.) 


=m 


Make,  a  =  a,     I  =  21^     and    ^i  =  12; 

Q       I      2'7 

then,  aS  =       "^        X  12  =  180. 

(3.) 

Make,  a  =  4,    I  =  20,    and    n  =  10. 

then,  S  =  (—2^)  X  10  =  120. 

(4.) 

S  =  (-^}  X  n. 

Make,  a  =  100,     5  =  200,     and    n  =  80. 

then,  S  =  (1^^°)  X  80  =  12000. 


66  KEY  TO  DAVIES'  NEW  ELEMENTAEY  ALGEBEA.        [269. 

(5.) 

Make,         a  =  500,     3  =  60,     and    n  =  20; 

/500  +  60\ 
then,  S  =  ( J- — j  X  20  =  5600. 

(6.) 

^  ==   (-2-)   ^  ^- 

Make,        a  =  800,     b  ~  1200,     and    9i  =-.  50. 

'800  +  1200\        ^^         ^^^^^ 
1  X  50  ==  50000. 


then,  S  =  \^ 


Page    267. 

(2.) 

I  —  a 

r  =  -• 

w  —  1 

Make,  ^  =  22,    a  =  4,    and    ?i  =  10; 

22  —  4         18 


then,  r  = 


10-1  9 


Page  269. 

(2.) 

z=  a  +  {n  —  l)r. 
Make,  a  =  2,    n  =  100,     and    r  =  7; 

then,       Z  =  2  +  (100  -  nV   ==   2  +  693   =  695. 


270. 1  ARITHMETICAL   PROGRESSION.  67 

(3.) 

First,  to  find  the  last  term.    We  have, 
I  —  a  -\-  {n  —  l)r ; 
and,  making  a  =  1^     ti  =  100,     and    r  =  2, 
we  have,     1=1  +  (100  -  1)2  :=   1  +  198  =:  199; 

then,     S  =  (^^)  xn  =  ^—   X  100  =  10000. 

(4.) 
To  find  the  last  term.     We  have, 

I  =  a  —  (n  —  l)r\ 

and,  making    a  ==  '70,    ;z  =  21,    and    r  =  3,    we  have, 

^  =  70  -  (21  -  1)  X  3   =  70  —  60  =  10; 

then,  8  =  \     T    )  X  ^^l 

and,  making    a  =  70,     ^  =  10,     and    n  =  21, 

we  have,  S  =   (— ~^)  X  21   =  840. 

(5.) 
To  find  the  last  term.     We  have, 

I  =  a  +  {n  —  l)r; 
and,  making     a  =  4,     ^  =  8,     and    r  =  8, 
we  have,      ^  =  4  +  (8  —  1)8  =  4  +  56  =  60; 

then,  S  =  \     T    )  X  n; 

and,  making    a  =  4,     ^  =i  60,     and    ^  =  8, 

we  have  S  =  i—^)  X  8  =  256. 


68  KEY  TO  DAVIES'  NEW  ELiaiENTARY  ALGEBKA.       [270. 


(6.) 

b  —  a 

T  =  -. 

n  —  1 

Make,  b  =  20,     a  =  2,     and    n  =z  10, 

,         ,                           20  —  2         18 
and  we  have,        r  =  -— =  —  =  2. 


b  —  a 
r  =  . 

n  +  1  . 

Make,  5  =  19,     a  =  4,     and    n  =  4; 

.-u  1.  19  -  4         15 

then  we  have,      r  =  — - —  =z  -—  —  Si 
5  5 

hence,  4. 7. 10. 13. 16. 19,     form  the  series. 

(8.) 
First,  to  find  the  last  term.     We  have, 
I  =  a  —  {71  —  l)r. 

Make,  a  =  10,     n  =  21,     and    r  —  ^\ 

then,        I  =:   10  -  (21  -  l)\  ==   10  -  6|  =  3i; 

then, 

a       /10  +  3J-\      ^^       30,10      ^^        40      ^,       840       ,,^ 

2 
(9.) 
We  have  the  equations, 

S  =  y  I  X  ^^,     and     I  =z  a  +  {7i  —  l)r. 


270.]  AEITHMKTTCAL    PKOGRKSSION.  69 

In  these  equations  all  the  quantities  are  known,  except  a 
and  I,  Substituting  the  numbers  for  the  known  quantities, 
we  have, 


2945 


=  (^-H!yx  n,    and     185  =  a  +  {n  -  1)6. 


( 10.) 
We  have,  from  Art.  183, 

h  —  a 

m  +  1 

Making    5  =  5,     and    a  =:  2,     and    m  =  9, 

5  2 

we  have,  r  =  =  0.3  ; 

from  which  the  terms  are  easily  found. 

(11.) 
We  have  the  formula, 

S  =  (-f-)  X  n. 

Now,   (^  =  1,   and  I  =:  n;  hence, 

(1  +  n\  (n  +  1\ 

^=(-2-)><^^  =  H-2-)- 

(12.) 
The  formula  for  the  last  term, 

I  =  a  -{-  {n  —  1)  X  r. 
'Making  a  =  1,   and  r  =  2,   we  have, 

Z  =   1  +  (?i  —  1)2  =:   1  +  2?2  —  2   =  2/^  —  1. 
Then,  in  the  formula, 


70  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.        [285. 

substitute  for  Z,  its  value,  and  for  a,  its  value,  1,  and  we  have, 

^        n  +  2n  —  1\ 

/3  =  I 1  X  n  =  n  X  n  zzz  n^. 

(IS.) 

In  this  example,  we  know  that  the  person  must  travel 
four  yards  to  place  the  first  stone  in  the  basket,  and  that  he 
must  travel  four  yards  in  addition  for  each  additional  stone 
which  he  brings.  Hence,  we  have  the  first  term,  the  com- 
mon difference,  and  the  number  of  terms,  to  find  the  sum 
of  the  series. 

First,  to  find  the  last  term.     We  have, 

I  =  a  -{-  {n  —  l)r. 

Making        a  =  4,     ^  =  100,     and     r  =  4, 

we  have,     1  =  4=  +  (100  —  1)4  =  4  +  396  =  400; 

then, 

S  =  (^^  xn  =  (^— )  X  100  =  20200  yards, 

which,  divided  by  1760,  the  number  of  yards  in  a  mile,  gives 
11  miles  and  840  yards. 

Page   285. 

(6.) 

Here  we  have  given  the  first  term,  the  common  ratio,  and 
the  number  of  terms,  to  find  the  last  term. 

Hence,  i:  =  1  x  2^  =  1  x  512  =  512    cents. 

Page   287. 

(4.) 
In  this  example,  we  have  the  first  term,  the  common  ratio, 


288.]  GEOMETRICAL    PROGRESSION.  71 

and  the  number  of  terms  given,  to  find  the  last  term  and 
the  sum  of  the  series. 

^  =   1   X  2^1  =   1   X  2048   =   2048. 

Then,  to  find  the  sum  of  the  series,  we  have, 

^  ^  Iq  -  a 
q  -1' 

in  which    I  =  2048,     q  =  2,    and    a  =  1; 

4096  -  1  ,^^, 

hence,  S  = =  4095. 

(5.) 

In  this  example,  we  have  given  the  first  term,  the  common 
ratio,  and  the  number  of  terms,  to  find  the  sum  of  the  series. 

First,  to  find  the  last  term,  we  have, 

Z  =  1  X  2"   =  2048. 

r^,            rY        Iq  —  a         4096  —  1  ,^^^    ,  .,,. 

Then,     S  =  -^ r= •   =  4095  shilhngs, 

which  is  equal  to  £204  155. 

(6.) 

In  this  example,  we  have  the  first  term,  the  ratio,  and  the 
number  of  terms,  to  find  the  last  term  and  the  sum  of  the 
series.    We  have, 

l:=zlxZ^  —  IX  19683   =:  19683  cents. 

^        Iq  —  a         19683  x  3  —  1         ^^^^, 

S  = = =  29524  cents 

9  —  1  2 

In  this  example,  we  have  the  first  term,  the  ratio,  and  the 
number  of  terms,  to  find  the  last  term  and  the  sum  of  the 
series. 


72  KEY  TO  DAVIES'  NEW  ELEMENTARY  ALGEBRA.       [289. 

I—  4  X  8^5  r=  4  X  35184372088832   -   140737488355328. 

For  the  sum  of  the  series,  we  have, 

140737488355328  X  8  —  4 
^=   • 8-~I 

that  is,  S  —  160842843834660 

(3.) 
First,  to  find  the  last  term,  we  have, 
I  =  aq^j 
and  making    a  =  512,     and    q  —  ^^    we  have, 

1 


^  =  512  X  HY  ==  512  X 


1024 


rrn  ^  a  —  ZO'  512   —  -i 

Then,  S  = ^  = — ^  =  682^. 

I  -  q  i  2 

(4.) 
First,  to  find  the  last  term,  we  have, 

I  —  aq^  =  2187  X  (iY  =  3. 

Then,         'S  =  ^^  =  ^-^   =  3279. 

(5.) 
First,  to  find  the  last  term,  we  have, 

I  =  aq^  =  972  X  (i)^  =  4. 
Then,  g^a-lq  _^12-^  _  ,,,, 


Then,      S 


1  — 

!Z 

(6.) 

I    zz 

:  aq'^ 

= 

147456  X 

{\r 

=   9. 

a 

-Iq 



147456  - 

±  ^ 

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By   OHAKLES    DAYIES,   LL.D. 

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This  work  is  not  intended  as  a  treatise  on  any 
special  branch  of  mathematical  science,  and  there- 
fore demands  for  its  full  appreciation  a  general  ac- 
quaintance with  the  leading  methods  and  routine  of 
mathematical  investigation.  It  is  an  elaborate  and 
lucid  exposition  of  the  principles  which  lie  at  the 
foundation  of  pure  mathematics,  and  a  highly  inge- 
nious application  of  their  results  to  the  development 
of  the  essential  idea  of  Arithmetic,  Geometry,  Al- 
gebra, Analytical  Geometry,  and  the  DifTerential  and 
Integral  Calculus.  The  work  is  preceded  by  a  gen- 
eral view  of  the  science  of  Logic,  and  closes  with  an 
essay  on  the  utility  of  Mathematics. 


RECOMMENDATIONS. 

We  are  very  much  mistaken  if  this  work  shall  not  prove  more  popular  and 
more  useful  than  any  which  the  distinguished  author  has  given  to  the  public. — 
Lutheran  Observer. 

We  have  been  much  interested  both  in  the  plan  and  in  the  execution  of  the 
work,  and  would  recommend  the  study  of  it  to  the  theologian  as  a  discipline  in 
close  and  accurate  thinking,  and  in  logical  method  and  reasoning.  It  will  be 
useful  also  to  the  general  scholar  and  to  the  practical  mechanic.  Nowhere  have 
we  seen  a  finer  illustration  of  the  connection  between  the  abstractly  scientific 
and  the  \>v»,cX\ca\.— Independent. 


A.  S.  Barnes  &  Burros  Publications. 


%n\t)xm'  "§\)ixm\s. 


By  EDWARD  D.  MANSFIELD. 

320  pp.  12mo.,  cloth.     Price  $1.26. 

This  work  is  suggestive  of  principles,  and  not  in- 
tended to  point  out  a  course  of  studies.  Its  aim  is 
to  excite  attention  to  what  should  be  the  elements 
of  an  American  Education  :  or,  in  other  words,  what 
are  the  ideas  of  a  Republican  and  Christian  Educa- 
tion, in  this  period  of  rapid  development.  The  author 
has  aimed  to  turn  the  thoughts  of  those  engaged  in 
the  direction  of  youth  to  the  fact  that  it  is  the 
entire  soul,  in  all  its  faculties,  which  needs  educa- 
tion, and  not  any  one  of  its  talents  ;  and  that  this  is 
a  need  especially  of  our  country  and  times.  To  do 
this  requires  a  complete  discipline  of  mind  and  analy- 
sis of  society. 


KECOMMENLATIONS. 

The  author  could  not  have  applied  his  pen  to  the  production  of  a  book  upon  a 
subject  of  more  importance  than  the  one  he  has  chosen.  His  views  upon  the 
elements  of  an  American  education,  and  its  bearings  upon  our  institutions,  are 
sound,  and  worthy  the  attention  of  those  to  whom  they  are  piuticuhuly  a^- 
dressed.— ifoc7ies<cr  Daily  Advertiser. 

We  have  examined  it  with  some  care,  and  are  delighted  with  it.  It  discusses 
the  whole  subject  of  American  education,  and  presents  views  at  once  enlarged 
and  comprehensive  ;  it,  in  fact,  covers  the  whole  ground.— Jcjr/.-.von  Pati  iot. 

We  hope  its  sentiments  maybe  diffused  as  freely  and  as  widely  thiou}j;lK)Ut  our 
land  as  the  air  we  hreathQ.— Kalamazoo  Gazette. 


A.  S.  Barnes  &  Burros  Publications. 


^^utlxm'  pbwtt*y. 


'ini^imittti 


By   E".  AV.  TAYLOR   ROOT. 

225  pp.  12mo.,  cloth.    Price  $1.25. 

This  work  has  origmatecl  in  the  acknowledged  prin- 
ciple that  improvement  of  the  mind  is  the  sure  at- 
tendant of  that  of  the  body.  It  contains  practical 
instruction  to  teachers  how  to  combine  pleasure  with 
profit,  so  as  to  render  them  mutually  subservient  to 
each  other. 


RECOMMENDATIONS. 

We  are  glad  to  welcome  this  new  contribution  to  the  cause  of  physical  educa- 
tion. The  introduction  of  proper  and  systematic  physical  exercises  to  the  school- 
grounds  will  not  only  save  many  lives,  but  will  greatly  aid  in  the  government 
and  moral  instruction  of  our  schools.— J/jcZu'.^au  Journal  of  Education. 

A  work,  completely  mi  generis,  is  School  Amusements,  by  N.  W.  Taylor  Root, 
a  gentleman  who  holds  to  the  good  old  plan  that 
"  All  work  and  no  play 
Makes  Jack  a  dull  boy." 
His  plan,  as  set  forth  in  this  book,  is  to  make  the  exercise  of  mind  and  body  go 
on  together. — Philadelphia  Press. 

A  capital  book  for  teachers  of  all  kinds  of  schools,  whose  design  is  to  show 
the  best  manner  of  intermingling  military,  gymnastic,  and  other  exercises  with 
the  duller  portion  of  school  duties.— ITarpe?-'*  Weeldy. 

A  new  influence  soems  to  be  creeping  over  our  school-houses,  and  we  must 
confess  to  a  hearty  sympathy  with  it.  Old  fogyism,  the  pedagogueship  of  the 
past,  with  its  rod  of  iron  or  its  black-hole,  and  the  rigid  sternness  of  the  master, 
is  at  last  relaxing  into  a  smile  at  its  own  absurdity  and  inhumanity,  and  is 
actually  endeavoring  to  make  school-days  pleasant  by  artificial  means.  A 
pleasant  school-room,  a  pleasant  teacher,  and  a  pleasant  crowd  of  "nappy  chil- 
dren !  Who  would  believe  it?  But  so  it  is,  and  so  it  ought  to  be,  and  the  author 
of  this  book  has  undertaken  to  reduce  school  amusements  and  pastimes  to  an  art 
worthy  of  study  and  cultivation.— i/MSicaZ  World. 


A.  S.  Barnes  S  Burr's  Publications. 


^tMUm'  ^iftravjj. 


OF 

By    IKA    MAYHEW,    A.M. 

474  pp.  12mo.,  cloth.     Price  $1,25. 

The  author  has  aimed  to  present  the  subject  of 
Education,  which  should  have  reference  to  the  whole 
man, — the  body,  the  mind,  and  the  heart, — and  so  to 
unfold  its  nature,  advantages,  and  claims,  as  to  make 
it  everywhere  acceptable.  He  would  have  a  good 
common  education  considered  as  the  inalienable 
right  of  every  child  in  the  community,  and  have  it 
placed  j^rs^  among  the  necessaries  of  life. 


RECOMMENDATIONS. 

We  commend  tlie'work,  not  merely  as  a  useful  manual  for  teachers  and  school 
committees,  but  as  one  to  be  read  by  the  people— every  man,  woman,  and  child 
of  -whom  is  interested  in  the  subject  of  which  it  treats.— Methodist  Quarterly 
Beview. 

This  is  truly  a  national  work,  and  should  be  in  the  hands  of  every  educator 
throughout  the  land.  We  can  not  speak  too  highly  of  it,  and  earnestly  recom- 
mend it  to  the  careful  perusal  of  all  interested  in  the  educational  reforms  of  the 
day.— Teachers'  Advocate. 


10 


A.  S.  Barnes  &  Burros  Publications. 


%i\w\\m'  pto^H* 


By  OHAELES   N0ETHE¥D,  A.M. 

327  pp.  12mo.,  cloth.     Price  $1.26. 

In  the  preparation  of  this  volume  it  has  been  the 
author^s  aim  to  furnish  for  teachers  a  work  that 
should  at  once  lead  them  to  view  their  calling  in  its 
true  light,  stimulate  them  to  fidelity,  and  furnish 
them  with  such  plain  practical  suggestions  as  might 
prove  valuable  td  them  in  the  performance  of  their 
important  and  arduous  duties.  In  the  execution  of 
his  design,  he  has  been  free  to  make  extracts  from 
the  writings  of  others,  when  he  has  found  their  views 
in  accordance  with  his  own.  In  all  such  cases  he 
has  made  the  proper  acknowledgment ;  and  it  is  be- 
lieved that  the  quotations  he  has  made  will  not  dimin- 
ish the  value  of  the  work. 


SECOMMENDATIONS. 

This  is  an  exceedingly  valuable  volume.  No  person,  parent,  teacher,  or  child, 
can  read  it  without  benefit.  It  is  a  treatise,  eminently  practical,  upon  Common 
School  Education,  and  it  is  full  of  wisdom,  evidently  gained  from  experience,  as 
well  as  from  a  careful  comparison  of  the  suggestions  of  other  minds.  He  has 
illustrated  almost  every  page  of  his  work  with  instances  and  examples  from  real 
life,  BO  apt  and  interesting  that,  even  merely  as  a  book  for  general  readers,  the 
work  has  great  attractions.  The  whole  spirit  and  tone  of  the  volume  is  admir- 
able.— Neio  Yoric  Evangelist. 

Mr.  Northend's  book  will  prove  interesting  to  all,  and  of  great  benefit  to 
teachers,  especially  as  a  chart  for  those  just  commencing  to  engage  in  the  profes- 
sion. We  are  glad  to  find  that  the  author,  in  furnishing  to  teachers  so  useful  a 
book,  has  not  neglected  the  sua-:iter  in  modo,  and  has  here  and  there  thrown  in 
a  pleasant  anecdote,  which  will  enliven  its  character  and  make  it  all  the  mor« 
acceptable.  —Massachusetts  Teacher. 


A.  S.  Barnes  &  Burros  Publications, 


§i09t^;iirUy. 


By  ALLAN  OUNKIN"GHAM. 

mi'\i\i  a  portrait  on  ^kcU 
286  pp.  12mo.,  cloth.      Price  $1.25. 

This  is  a  candid,  truthful,  and  appreciative  memoir 
of  the  great  painter,  with  a  compilation  of  his  various 
discourses,  which  are  possessed  of  so  much  real  merit 
as  to  entitle  him  to  almost  as  much  credit  as  a 
writer  and  orator  as  a  painter.  His  observations  in 
relation  to  drawing  and  painting  are  as  valuable  now 
as  when  they  were  uttered.  The  volume  will  always 
be  a  text-book  with  artists,  and  may  be  read  with 
advantage  by  all  who  are  attempting  to  acquire  the 
rudiments  of  the  art. 


EECOMMENDATION. 

This  volume  is  at  once  a  book  of  pleasant  reading,  biographical  and  critical, 
and  an  educational  guide  for  the  uninitiated  in  the  art  of  painting  ;— the  annual 
discourses  of  the  prince  of  English  portrait-pain ers,  pronounced  before  the  ilite 
of  British  art  in  the  Royal  Acaderaj',  embodying  criticisms  upon  the  great  mas- 
ters of  painting  and  the  principles  of  the  art,  which,  however  modified  by  later 
criticism,  are  of  the  highest  value,  not  only  to  artists,  but  to  all  who  aspire  to  b© 
connoisseurs.  The  readers  of  Ruskin's  imperious  criticisms  should  by  all  means 
acquaint  themselves  with  the  plain,  practical,  common-sense  judgment  of  Sir 
Joshua  Reynolds. — Independent. 


A.  S,  Barnes  &  Burr's  Fuhlications. 


%\\\mi\\  3ixu%. 


K.  G.  PARKER  and  J.  M.  WATSON. 


NATTONALi  1E*RIMER,  or  WORD-BUILDER 

NATIONAL  ELEMENT AR"R    SPELLER 

NATIONAL   PRONOUNCING    SPELLER 

For  these  books  the  publishers  ask  only  one  favor, 
that  teachers  and  school-officers  would  examine  their 
merits  before  adopting  other  works.  The  word- 
method  and  dictation  exercises,  prepared  with  great 
skill  and  care,  form  prominent  features. 


RECOMMENDATIONS. 

The  new  Spelling-book  is  a  crowning  excellence  to  the  series.  It  is  destined 
to  sui>plant  Noah  Webster's  in  popularity. — Samuel  P.  Bates,  Deputy  Superin- 
tendent Common  ScJiools  of  Pennsylvania. 

This  book  is  far  in  advance  of  any  similar  work  with  which  we  are  acquainted. 
After  a  critical  examination,  we  are  happy  to  recommend  it  to  teachers  and 
school-officers,  as  being  every  way  worthy  of  the  prominent  position  the  Spelling- 
book  ought  to  occupy  in  primary  instruction. — New  York  Teacher. 

This  is  one  of  the  most  complete  works  with  which  we  have  met.  The  arrange- 
ment of  words,  and  the  systematic  classification  in  its  preparation,  can  not  fail  to 
accomplish  the  vrork  designed. — Prof.  F.  A.  Allen,  Principal  of  Normal  School, 
Westchester,  Pa. 

I  have  examined  the  National  Pronouncing  Speller,  and  am  free  to  say  that  it 
is  the  only  work  I  ever  saw  which  exactly  meets  my  idea  of  what  a  Spelling-book 
should  be.  It  is  pre-eminently  practical ;  it  requires  the  child  to  do  what  it  will 
be  necessary  for  him  to  do  all  through  his  life  ;  it  requires  him  not  only  to  tell 
how  the  letters  are  arranged  to  fo'-m  the  word,  but  to  write  it,  using  it  according 
to  its  signification.  It  needs  only  to  be  seen  and  understood  to  meet  with  favor.— 
If.  R.  Barnard,  Prin.  Union  School,  Ithaca,  N.  T. 


A.  S.  Barnes  &  Burr's  Publications. 


§atio»at  ^tx'm. 


BY 

K.  G.  PARKER  and  J.  M.  WATSON. 


NATIOJVAli  FIRST  READER,  118  pp $ 

NATIONAL  SECONO  READER,  ^^4:  pp 

NATIONAL!  THIRD  READER,  ^88  pp 

NATIONAL  FOURTH  READER,  405  pp 

NATIONAL.  FIFTH  READER,  600  pp 

This  series  of  Readers  is  unsurpassed  bj  any  ever 
issued  from  the  American  press,  in  the  excellence  of 
its  selections,  in  the  proper  grading  of  the  pieces,  in 
its  admirable  system  of  elocution,  in  the  variety  and 
interest  of  the  matter  appended  in  the  form  of  notes, 
and  in  the  substantial  and  beautiful  style  of  art  in 
which  they  are  published. 


BECOMMENDATIONS. 

These  Readers,  in  my  opinion,  are  the  best  I  have  ever  examined.  I  have  had 
better  success  witu  my  reading-classes  since  T  commenced  training  them  on  these, 
than  I  ever  met  with  before. — A.  P.  Harrington,  Principal  of  Union  School, 
Marathon,  iV.  Y. 

In  the  simplicity  and  clearness  •with  which  the  principles  are  stated,  in  the  ap- 
propriateness of  the  selections  for  reading,  and  in  the  happy  adaptation  of  the 
different  parts  of  the  series  to  each  other,  these  works  are  superior  to  any  other 
text-books  on  this  subject  which  I  have  examined.— C7iar?ea  S.  Halsey,  Prin. 
Collegiate  Institute,  Newton,  JV.  J. 

From  a  brief  examination  of  them,  I  am  led  to  believe  that  we  have  none  eiiual 
to  them.  I  hope  they  will  prove  as  popular  as  they  are  excellent. — Pro/.  Fred. 
S.  Jewell,  N.  Y.  State  Normal  School. 

The  National  Readers  and  Speller  I  have  examined,  and  carefully  compared 
with  others,  and  must  pronounce  them  decidedly  superior,  in  respect  to  literary 
merit,  style,  and  price. — iV^.  A.  Ilamilton,  Pres.  Teachers^  Union,  Whitewater,  Wis. 

I  consider  them  emphatically  the  Readers  of  the  present  day,  and  T  believo 
that  their  intrinsic  merit  will  insure  for  them  a  full  measure  of  popularity,— 
J.  W.  Schertnerhorn,  Prin.  Coll.  Institute,  Middletoton,  N.  J. 


A.  S.  Barnes  &  Burr'st  Publications. 


^ixtimnl  ^»m^. 


#tt|Iirt  « 


By  S.  W.  OLAKK,  A.M. 

Clark's  First  Lessons  in  English  Grammar $ 

Clark's  New  English  Grammar 

Key;  to  Clark's  Grammar 

Analysis  of  the  English  Language 

Grammatical  Chart 

The  true  place  to  test  any  text-book  is  the  class- 
room. Tried  by  this  test,  Clark^s  Grammars  have 
won  unqualified  commendation.  Pupils  become  in- 
terested in  the  study  more  readily  and  generally  by 
this  than  by  any  other  system.  The  reason  is  ob- 
vious. Any  species  of  instruction  that  can  be  ad- 
dressed to  the  eye  is  more  easily  and  quickly  seized 
than  when  presented  in  any  other  way.  The  system 
of  diagrams,  with  the  use  of  the  blackboard,  makes  it 
both  easy  and  philosophic. 


RECOMMENDATION. 

Clark's  Grammar  is  a  new  thing  in  the  study  of  langnage  ;  by  his  system,  the 
blackboard,  the  great  weapon  of  the  modern  educator,  is  made  to  play  an  im- 
portant part  even  in  the  ordinarily  dry  and  dull  study  of  English  Grammar. 
His  diagrams  are  at  once  simple  and  unique  in  conception,  and  universal  in 
application.  The  most  wild  and  uncouth  sentences  that  Carlyle  ever  wrote, 
equally  with  the  most  polished  and  njellifluous  of  Byron  or  Tom  Moore,  are 
readily  caught,  tamed,  and  made  to  trot  in  double  or  single  harness  through  the 
scholar's  parsing  vocabulary.  While  looking  over  the  pages  of  Clark's  New 
Grammar,  we  could  not  but  think  of  old  Lindley  Murray,  and  the  aching  heads 
that  used  in  our  boyish  days  to  pore  over  his  crabbed  pages,  at  d  wonder  why 
nobody  thought  of  so  obvious  an  improvement  before. — Racine  Advocate. 


A.  S.  Barnes  &  Burr's  Publications. 


§niimu\  3tn6^- 


BY 

JAMES  MON^TEITIT  and  FRAKCIS  McN"ALLY. 


Monteith's  First  Lessons  in  Geography i 

Monteith's  Introduction  to  Manual  of  Geography 

Monteith's  Wew  Manual  of  Geography 

McWally's  Complete  School  Geography 


RECOMMENDATIONS. 

All  the  geographies  in  use  in  our  common  schools  have  received  from  ir.e  a 
careful  and  critical  examination.  The  National  Series  was  one  of  two  series 
that  received  my  full  approbation.  The  opinion  that  I  formed  of  their  gieat 
merit  is  justiGed  by  their  extensive  use  in  the  public  schools  of  this  city.  I  hHve 
found,  by  examination  of  the  Book  of  Supply  of  our  Board,  that  considerably  the 
largest  number  of  any  series  now  used  in  our  public  schools,  is  the  National,  by 
Monteith  and  McNally. — B.  A.  Adams,  Chairman  of  ^'^  Conimittee  on  Course  of 
Studies  and  School-hooJfs,^'  and  Member  of  "  Committee  of  Supplies"  of  Board  of 
Education  of  New  York. 

During  an  experience  of  ten  years  in  teachinsj,  I  have  found  no  series  of 
Geographies  so  well  calculated,  in  matter,  arrangement,  and  system,  to  facilitate 
the  progress  of  the  learner  as  Monteith  and  McNally's.— Solomon  ili/erl,  Prof. 
of  English  Grammar  and  Geography  in  the  York  Co.  Normal  School,  Pa. 

This  s;nies  was  adopted  after  a  careful  examination  of  the  best  works  in  this 
branch  of  study,  and  a  year's  experience  makes  us  better  and  better  satisfied  with 
our  choice. — Josiah  T.  Read,  Priii.  JIarshall  (Mich.)  Union  SchooL 

We  have  used  McXally's  Oeogrnphy  since  its  publication,  deetning  it  the  best 
class-book  in  the  market.  It  not  only  is  the  equal  of  its  rivals  in  positive  merits, 
bnt  is  superior  to  most  of  them  as  to  icJiat  it  omits.  It  is  both  practical  and  prac- 
ticahle  as  a  text-book. — Jiev.  Joseph  E.  King,  A.M.,  Principal  of  Fort  Edward 
Institute,  N.  Y. 

We  have  used  McNally's  and  Monteith's  Geographies  for  three  years,  and 
would  not  exchange  them  for  any  others  In  the  maiKet. — Rev.  B.  St.  James  Fry, 
A.  M.,  President  of  Worthington  Female  College,  Ohio. 


A.  S.  ISarnes  <£•  Burr's  Publications. 


ptioual  3ixiu. 


DAVIES' 
DAVIES' 
DAVIES' 
DAVIES' 
KEY  TO 
DAVIES' 
KEY  TO 
DAVIES' 
DAVIES' 
KEY  TO 
DAVIES' 
DAVIES' 

DAVIES' 
KEY  TO 
DAVIES' 
KEY  TO 
DAVIES' 
DAVIES' 
DAVIES' 
DAVIES' 
DAVIES' 
DAVIES' 
DAVIES' 
DAVIES' 
DAVIES' 


By  CHARLES  DAVIES,  LL.D. 

Elementarg  (Course. 

PRIMARY  ARITHMETIC  AND  TABLE-BOOK $ 

FIRST  LESSONS  IN  ARITHMETIC 

INTELLECTUAL  ARITHMETIC 

NEW  SCHOOL  ARITHMETIC 

DAVIES'  NEW  SCHOOL  ARITHMETIC 

NEW  UNIVERSITY  ARITHMETIC 

DAVIES'  NEW  UNIVERSITY  ARITHMETIC... e         

GRAMMAR  OF  ARITHMETIC 

NEW  ELEMENTARY  ALGEBRA 

DAVIES'  NEW  ELEMENTARY  ALGEBRA 

ELEMENTARY  GEOMETRY  AND  TRIGONOMETRY 

PRACTICAL  M ATHEM ATICS 

^Ttj&anceti  Louise, 

UNIVERSITY  ALGEBRA 

DAVIES'  UNIVERSITY  ALGEBRA 

BOURDON'S  ALGEBRA 

DAVIES'  BOURDON'S  ALGEBRA 

LKGENDRE'S  GEOMETRY 

ELEMENTS  OF  SURVEYING 

ANALYTICAL  GEOMETRY 

DIFFERENTIAL  AND  INTEGRAL  CALCULUS 

DESCRIPTIVE  GEOMETRY 

SHADES,  SHADOWS,  AND  PERSPECTIVE 

LOGIC  OP  MATHEMATICS 

MATHEMATICAL  DICTIONARY 

Mathematical  Chart  (Sheet) 


EECOMMENDATIONS. 

We  have  tested  the  completeness  of  tliis  dictionary  by  looking:  for  a  consider- 
able number  and  variety  of  titles,  under  all  of  which  we  have  found  statements  and 
discussions,  succinct  without  being  obscure,  and  sufficiently  thorough  to  render  the 
work  a  reference-book  for  proficients  as  well  as  for  pupils  in  Mathematics.  It  is 
just  such  a  book  as  we  have  needed  for  a  score  of  years. — North  Amer.  Review. 

Each  treatise  serves  as  an  introduction  to  the  next  higher  by  the  similarity  of 
its  reasonings  and  methods,  and  the  student  is  carried  forward  by  easy  and 
gradual  steps  over  the  whole  field  of  mathematical  inquiry,  and  that,  too,  in  a 
shorter  time  than  is  usually  occupied  in  mastering  a  single  department.  I  sin- 
cerely and  heartily  commend  them  to  the  attention  of  ray  fellow-teachers  in 
Canada. — John  McLean  Bell,  B.  A.,  Prin.  Lower  Canada  College. 

The  undersigned  has  examined  with  care,  and  taught  some  time  since,  several 
volumes  of  Davies'  Mathematics,  and  is  of  the  opinion  that,  as  a  whole,  it  is  the 
most  complete  and  best  course  for  academic  and  collegiate  instruction  with  which 
he  is  acquainted. — Horace  Webster,  LL.D.,  Fres.  N.  Y.  Free  Academy. 


A.  S,  Barries  S  Burros  Publications. 


i»ti0ma  Mxm. 


tm 


By  EMMA  WILLARD. 


School  History  of  the  United  States $0.75 

Large  History  of  the  United  States 1.50 

The  author^s  long  experience  as  a  teacher  at  the 
head  of  one  of  the  most  noted  Seminaries  of  the 
country,  gave  her  great  advantage  in  preparing  a 
book  on  our  National  History  adapted  to  use  in 
schools.  The  unqualified  commendation  bestowed 
upon  these  books  by  the  leadmg  statesmen  of  the 
age  is  a  sufficient  guarantee  of  their  merit. 


EECOMMENDATIONS. 

I  can  not  better  express  my  sense  of  the  value  of  your  History  of  the  United 
States,  than  by  saying  I  keep  it  near  me  as  a  book  of  reference  accurate  in  facts 
and  dates. — Daniel  Webster. 

Similar  testimonials  have  been  received  from  Henry  Clay,  John  McLean, 
Elijah  Willard,  Senator  Dickinson  and  many  other  eminent  men. 

This  is  a  noble  and  well-'w'ritten  book,  whi;h  it  is  both  a  pleasure  aiid  a  profit 
to  read.  The  sty  .6  is  lucid,  and  varies  with  the  impulse  of  the  subject.  Mrs. 
Willard  should  1  e  considered  as  a  benefactress,  not  only  by  her  own  sex,  of 
whom  6h«  became  in  early  years  a  pyorainent  and  permanent  educator,  but  by 
the  country  at  large,  to  whose  good  she  has  dedicated  the  gathered  learning  and 
faithful  labor  of  life's  later  periods.  The  truths  that  she  has  recorded,  and  the 
principles  that  she  has  impressed,  will  win  from  a  future  race  gratitude  that  can 
not  grow  old,  and  a  garland  that  will  never  fade.— Jfrs.  Z.  //.  Sigoumey. 


18 


A.  S.  Barnes  &  Burros  Puhlicaiions. 


1.  Principles  of  Chemistry,  embracing  the  most  recent 

discoveries "  in  the  Science,  and  the  outlines  of  its 
application  to  Agriculture  and  the  Arts.  Illustrated 
bj  numerous  experiments  newly  adapted  to  the  sim- 
plest apparatus.  By  John  A.  Poetek,  A.  M.,  M.  D., 
Professor  of  Agriculture  and  Organic  Chemistry  in 
Yale  College.  Price  $1.25. 
[A  Box  OF  Apparatus  prepared  expressly  for  this  Work,  will 

COST  ONLY  $8.00.] 

2.  First  Book  of  Chemistry,  and  Allied  Sciences,  in- 

cluding an  outline  of  Agricultural  Chemistry,     by 
Prof.  John  A.  Poetee.     Price  60  cents. 


RECOMMENDATIONS. 

I  am  greatly  pleased  with  Porter's  Chemistry.  It  does  not  give  a  mere  outline, 
but  exhibits  the  atomic  proportions  in  all  combinations  so  as  to  make  the  student 
scientific  in  his  processes.  It  is  fully  up  with  the  progress  of  the  age  in  this  most 
rapidly  advancing  science  ;  and  last,  not  least,  in  its  behalf,  it  is  so  simple,  easy, 
and  beautiful,  that  a  teacher  can  make  himself  a  chemist  while  teaching  a  class. — 
Prof.  J.  Young,  of  the  North- Western.  ChrUtian  University. 

1  put  a  class  into  it  last  winter,  which  nearly  finished  it.  I  think  I  never  used  & 
text-book  on  any  subject  that  pleased  me  so  well.  It  seems  just  what  our  acade- 
mies require.  I  feel  sure  that  this  book  meets  a  great  and  general  want,  and  shall 
be  much  mistaken  if  it  does  not  prove  one  of  the  most  successful  ever  issued. — 
M.  Sfurges,  Monroe,  Butler  County,  Ohio. 

I  have  examined  Porter's  Chemistry  with  no  little  attention.  As  an  elementary 
Text-book  it  is  a  model.  The  bork  must  greatly  extend  the  study  of  this  important 
science.— JS.  A.  Grant,  Franlcfort,  Ky. 

I  do  not  hesitate  to  pronounce  it  a  work  most  admirably  adapted  to  convey  to 
any  inquiring  mind  a  clear  idea  of  the  elements  of  chemistry. — M.  L.  Morse,  Prin- 
cipal of  High  School,  Dover,  N.  H. 

We  have  read  this  book  with  great  pleasure,  and  can  recommend  it  as  the  sim- 
plest, most  concise,  and  most  comprehensive  School  Chemistry  known  to  us.  Its 
simplicity  is  its  great  va.QXit.— Massachusetts  Teacher. 


A.  S.  Barnes  &  Burr'' 8  Publications. 


"§'Ximm\  3txm. 


INTRODDCTORY  COUKSE  OF  NATURAL  PHILOSOPHY. 

FROM  THE  FRENCH  OF  M.  GANOT. 

By  professor  W.  G.  PECK,   M.  A. 
Price  $1.25. 

One  of  the  most  attractive  volumes  ever  published  is 
this  new  work  on  Natural  Philosophy.  The  original  edi- 
tion is  regarded  as  the  standard  text-book  in  France,  and 
emanates  from  one  of  her  ablest  and  most  popular  writers. 
It  has  been  Americanized  and  adapted  with  remarkable 
skill  to  the  peculiarities  of  our  systems  of  instruction,  by 
Professor  Peck.  One  of  the  most  striking  features  of  the 
book  is  its  profuse  and  magnificent  system  of  illustration. 
By  special  arrangement  with  M.  Ganot,  fac-similes  of  all 
the  original  engravings  are  presented,  rendering  it  un- 
equalled in  point  of  beauty,  accuracy,  abundance  and 
general  adaptation  of  illustration.  The  volume  embraces 
a  full  and  logical  development  of  all  the  elementary  prin- 
ciples of  Physics. 

RECOMMENDATIONS. 

I  have  examined  Peck's  Ganot's  Natural  Philosophy  with  some  care,  and  know 
not  which  to  admire  most,  the  lucid,  compact,  and  charming  style  of  the  composi- 
tion, or  the  beautiful  and  faultless  mechanical  execution  of  the  book.  The  typo- 
graphical execution  seems  to  be  unsurpassed,  and  the  illustrations  are  numerous, 
accurate,  and  elegant.  It  is  just  such  a  book  as  students  should  have  in  their  hands, 
as  a  model  of  excellent  scholarship  and  good  taste.  We  have  adopted  it  in  this 
Institution.— /rri  W.  AUetiy  Smtior  rrofeasor  of  Union  Chrintian  College. 

The  Natural  Philosophy  of  M.  Ganot,  edited  by  Prof.  Peck,  is,  in  my  opinion, 
the  best  work  of  its  kind,  for  the  use  intended,  ever  published  in  this  country.— 
D.  C.  Van  Norman,  Principal  of  the  Van  Norman  Inslitule,  New  York. 


A.  S.  Barnes  &  Burr's  Publications. 


%ixXm\\\  ^n'm. 


EMBRACING 

Grammar,  Conversation,  Literature,  and  Commercial  Cor- 
respondence, together  with  an  Adequate  Dictionary.     By 
Louis  Pujol,  A.  M.,  of  the  University  of  France,  and  D.  C. 
Van  Norman,  LL.D.,  of  N.  Y.  Young  Ladies'  Institute. 
Price  $1.50. 

Part  I.  inchides  an  elaborate  Treatise  on  Pronunciation;  u 
Theoretical  and  Practical  Grammar,  on  a  plan  entirely  new,  with 
Exercises  in  both  French  and  English,  and  a  series  of  Vocabula- 
ries, forming  an  adequate  dictionary. 

Part  IL  presents  a  complete  and  orderly  development  of  French 
Syntax,  with  Exercises  in  both  French  and  English,  and  Polite 
Conversations  ilhistrative  of  the  idiomatical  rules.  It  includes  also 
a  general  Course  of  Versions,  which  embraces  an  instructive  series 
of  biographies,  and  a  very  useful  abstract  of  the  history  of  France. 
^  Part  IlL  contains  a  new  system  for  teaching  French  Conversa- 
tion. The  exercises  embrace  a  great  variety  of  topics,  and  are 
illustrative  of  more  than  four  thousand  words  and  phrases  in 
common  use. 

Part  IV,  includes— 1.  Progressive  Lessons  in  Translating,  as 
an  introduction  to  French  Literature. — 2.  Selections  in  prose  and 
verse,  from  the  French  Classics  and  the  best  modern  writers. 
Also,  an  Appendix  of  Polite  and  Commercial  Correspondence, 
and  translations  of  all  the  more  difficult  words  and  phrases,  with 
figures  of  reference,  together  with  an  Adequate  Dictionary, 


RECOMMENDATIONS 

I  take  great  pleasure  in  recommending  Pujol  and  Van  Norman's  French  Class- 
book,  and  there  is  no  French  Grammar  or  Class-book  which  can  be  compared  with 
it  in  completeness,  system,  clearness,  and  general  utility. — Elias  Peissner,  Prof, 
of  German,  and  Lecturer  on  Political  Economy,  Union  College. 

I  have  been  examining  with  increasing  admiration  your  Course  of  Instruction  in 
the  French  L8,Tiguage.  It  furnishes  all  that  is  needful  to  qualify  the  student  not 
only  to  speak  and  write  the  French  with  fluency,  I  ut  also  to  read  any  French 
author.— i?eu.  A.  L.  Lindsey,  A. 31.,  Prin.  Parsonage  I;,st.,  South  Salem,  iV.  Y. 

The  plan  of  your  Complete  French  Class-book  is  as  ingenious  as  it  is  unique,  and 
appears  to  me  admirably  adapted  tojthe  purposes  of  instruction. — Bev.  W.  Ormistan, 
D.D.,  of  Hamilton,  Canada  West.    '" 

Having  been  engaged,  occasionally,  for  forty  years  in  teaching  the  French  lan- 
guage, I  have  ca»-efully  examined  nearly  all  the  works  on  the  subject  that  have 
been  published  in  this  country,  and  am  free  to  say  that  I  consider  thic,  on  the 
whole,  decidedly  superior  to  any  other.  I  have  already  adopted  it  in  my  class.— 
— A.  Curtis,  M.M.,  M.D.^  Pres.  Lit.  and  Sci.  Institute,  Cincinnati. 


A.  S.  Barnes  &  Burros  Publications. 


%^imxdX  ^nm. 


CLASS-BOOK   OF   BOTANY;    being  Outlines  of  the 
Structure,  Philosophy,  and  Classification  of  Plants,  with  a 
Flora  of  the  United  States  and  Canada.    By  Alphonso  Wood, 
A.M.,  Principal  of  Female  Academy,  Brooklyn. 
832  pp.  8vo.     Price  $2  25. 

Teachers,  pupils,  and  amateurs,  ^vill  find  in  the  new  Class-book 
of  Botany  the  following  peculiar  advantages  : 

First  The  Scientific  Treatise  embraced  in  Parts  I.,  II.,  and  III.,  is  re- 
markable for  its  comprehensiveness,  clearness,  and  brevity;  is  divided  into 
short  paragraphs,  suited  to  the  learner's  convenience,  and  each  paragraph, 
with  the  topic  prefixed  in  capitals,  is  adapted  to  recitation  and  reviews. 

Second.  The  Flora  comprehends  a  wider  territory  than  that  of  any  School 
Botany  now  in  use,  extending  from  the  Atlantic  to  the  Mississippi,  and 
from  the  St.  Lawrence  to  the  Gulf  of  Mexico. 

Third.  The  Flora  comprehends  not  only  our  native,  spontaneous  vege- 
tation, but  also  one  thousand  species  of  cultivated  plants — almost  our  entire 
exotic  flora,  growing  in  the  field,  the  garden,  and  the  conservatory;  thus 
rendering  the  study  of  Botany  as  practicable  in  the  city  as  in  the  country. 

Fourth,  The  descriptions  of  species  are  unusually  full,  graphic,  and 
popular  in  style. 

Fifth.  The  Tables  for  analysis  are  far  in  advance  of  those  in  the  former 
editions,  both  in  simplicity  and  extent,  reaching  now  from  the  Grand  Divi- 
sion to  the  Species,  and  by  thoir  peculiar  form  adapted  to  class  exercise 
throughout  the  entire  route ;  thus  rendering  the  pursuit  at  once  a  vigorous 
discipline  and  an  exciting  amusement 


RECOMMENDATION. 

Professor  Wood  is  well  known  among  Botanists,  and  it  wonld  be  presumption  in 
us  to  affect  to  praise  him.  His  Class-book  was  first  issued  fifteen  years  ago,  and 
has  had  extensive  sale  and  use.  It  is  now  thoroughly  revised  ;  we  might  say,  re- 
written. The  results  of  the  author's  repeated  and  far-extended  journeys  and  re- 
searches are  here  given.  Ue  says  that  most  of  the  descriptions  are  verified  by  his 
own  personal  observation.  The  list  of  plants  includes  every  species  known  to 
exist  from  the  St.  Lawrence  to  the  Gulf  of  Mexico,  and  from  the  Atlantic  to  the 
Mississippi,  and  also  the  exotics  which  have  obtained  any  considerable  culture  in 
this  country.  The  illustrations  are  abundant  (numbered  to  745),  and  well  cut. 
The  first  part  of  the  book,  nearly  200  pages,  treats  at  length  of  the  structure *of 
plants,  of  ibo  uses  of  their  parts,  and  of  classification.  The  information  given  is 
full,  and  the  style  is  clear  and  direct ;  with  excellent  tables,  index,  and  a  glossary. 
The  book  is  excellent,  both  in  substance  and  in  style. — lUinoia  Teacher. 


A.  S.  Barnes  &  Burr's  Publications. 


patwiral  Mnu. 


POR  TEACHERS  IN  PiaMAlU'  SCHOOLS  AND  rRIMARY  CLASSES. 
BY   A.  S.  WELCH, 

PRINCIPAL  OF  MIOrilGAN  STATE  NORMAL  SCHOOL. 


From  Pestalozzi: — ^''Observation  is  the  al)solute  lasts  of  all  knowl- 
edge. The  first  object,  then,  in  education^  must  be  to  lead  a  child  to 
observe  with  accuracy;  the  second^  to  express  with  correctness  the 
result  of  Jiis  observations.'^'' 

Principles,  without  application,  have  been  harbored  and  sus- 
tained by  many  of  the  so-called  disciples  of  Comenius  and  Pesta- 
lozzi, but  we  look  in  vain  for  a  development,  in  systematic  ob- 
servation, at  all  commensurate  with  the  practical  results  that 
might  have  been  expected  to  follow  the  teachings  of  these  two 
educators. 

In  commending  this  book  to  the  general  examination  of  teach- 
ers, the  publishers  would  remark,  that  the  author  brings  to  its 
preparation  great  and  varied  experience  in  our  schools,  and  ex- 
tended observation  in  those  schools  of  England  and  Germany 
where  a  systematic  presentation  of  the  subject  has  been  secured. 


The  following  Text-books  will  be  found  to  contain  principles 
involving  the  method  of  Obiect-teaching,  and  are  warmly  com- 
mended to  the  attention  of  educators  :  • 

J. — Object  Lessons  ;  designed  for  the  use  of  Teachers  in  Primary  Classes  and 
Primary  Schools.     By  A.  S.  Welch.    Price  50  cents. 

2.— Exercises  for  Dictation  and  Pronunciation  ;  embracing  a  numerous  col- 
lection of  difficult  words,  including  nearly  300  military  and  war  terms,  together 
with  a  variety  of  useful  lessons.     By  Charles  Northend.    Price  50  cents. 

3. — Juvenile  Definer  :  a  collection  and  classification  of  familiar  'words  and 
names,  correctly  spelled,  accented,  and  defined.  By  Wm.  W.  Smith.  Price 
30  cents. — The  words  are  grouped  with  referenc'e  to  similar  signification  or  use : 
as  the  several  kinds  of  huildings  compose  one  class,  vessels  another,  cloths 
another,  &c.,  &c. 

4.— First  Book  in  Composition.  By  F.  Brookfield.  Price  30  cents.— This 
little  book  is  a  successful  attempt  to  aid  thought,  by  a  series  of  illustrations  and 
suggestions  of  topics  calculated  to  inspire  interest  in  a  study  heretofore  repulsive 
to  a  child.  Subjects  have  been  selected  upon  which  the  thoughts  of  all  children 
exercise  themselves  spontaneously. 

5.— The  Child's  Book  in  Natural  History  ;  illustrating.the  Animal,  Vegetable, 
and  Mineral  Kingdoms,  with  application  to  the  Arts.  By  M.  M.  Carl.  Price 
35  cents, 

5. — Treasury  c/Knowledge  :  embracing  Elementary  Lessons  in  common  things! 
—Practical  Lessons  on  common  objects— Introduction  to  the  Sciences.  By  Wil- 
liam and  Robert  Chambers.  Price  75  cents.— This  last-named  book  contains 
a  vast  amount  of  information,  and  deserves  a  place  in  every  school-room. 


A.  S.  Barnes  &  Burr'^s  Publications. 


§ati0»»I  3mt^. 


BY  CHAKLES'NORTHEND,  A.M. 
250  pp.  18mo.    Price  50  cts. 

IsTo  work  has  been  prepared  for  teachers  and  scholars 
that  will  excite  more  interest  and  pleasure,  than  a  small 
book  by  Charles  ISTorthexd,  of  the  State  Normal  School 
of  Connecticut,  entitled  "Exekoises  for  Dictation  and 
Pronunciation;"  containing  a  large  numberof  the  most 
difficult  words  in  the  language,  including  nearly  three 
hundred  military  and  war  terms,  and  two  thousand  words 
which  are  frequently  mispronounced  as  well  as  misspelled ; 
alphabetically  arranged,  pronunciation  indicated,  and  mean- 
ing given ;  together  with  a  variety  of  other  useful  lessons. 


This  work  will  prove  a  valuable  Hand-book  for  both  teacher 
and  scholar,  and  should  have  a  place  in  every  school. 

It  has  not  been  the  aim  of  the  author  to  furnish  a  substitute  for 
the  spelling-book,  but  rather  to  prepare  an  accompaniment  to  it, 
for  the  use  of  higher  classes.  In  the  several  collections  or  group- 
ings of  words,  the  most  prominent  in  each  department  have  been 
brought  together.  The  various  miscellaneous  exercises  in  the 
book  will,  it  is  believed,  readily  commend  themselves  to  teachers, 
and  open  a  wide  field  for  nnich  general  instruction  in  every-day 
matters — making  a  good  basis  for  useful  Object  Lessons. 

The  following  table  of  contents  will,  perhaps,  best  exhibit  the 
main  features  of  the  work,  viz. : 

Hints  on  Spelling— Rnles  for  the  use  of  Capitals— Rules  of  Spelling— Words  simi- 
lar in  pronunciation,  but  dissimilar  in  spelling  and  meaning- Words  pronounced 
nearly  alike,  but  differing  in  spelling  and  dcUning — Words  of  two  pronunciations — 
Synonyms- Words  of  special  resemblance- Words  varying  in  use— Words  liable 
to  be  misspelled— Cluistian  names  of  males  and  females— Occupations— Professions 
— Auiraals—Birds— Fishes— Trees— Flowers— Productions  of  the  farm  and  garden — 
Agricultural  implements— Furniture  and  articles  of  household-Arithmetic— Geog- 
raphy—Grammar— Philosophy— Botany— Physiology— Forms— Books— Wearing 
apparel— Architective  origin  of  words — Military  and  government  terras — Hardware 
—Boats  and  parts  of  a  ship— Marine  journal— Review  of  tho  market— Monetary 
affairs— Exports— Prefixes— State  mottoes— Abbreviations— Proof-marks,  &c. 


-.'."l'*™"^^^  CENTS 

DAY    AND     TO     $Uo!    oTthbV^  ''°^'^rH 
OVERDUE.  "     ^"^E    SEVENTH     DAY 


"        MSmg^tfSr-- 


